Skip to content
← Mathematics

Mathematics · Class 11 Science

Ch 3Trigonometry — Class 11 Mathematics, concept-first.

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. The name itself comes from two Greek words — trigonon (triangle) and metron (to measure) — so trigonometry literally means "measuring triangles."

175

Q&A

8

Concepts

~7m

Unit weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Radian Measure

An angle is generated by rotating a ray (the initial side) about its endpoint (the vertex) until it reaches a final position (the terminal side).

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

3.1

Introduction

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles.

3.2

A recall of basic results

Until now, trigonometric ratios have been defined only for the acute angles of a right triangle, and the identities proved were valid only in that restricted setting.

3.2.1

Angles

An angle is formed by rotating one ray about a fixed point until it reaches a second ray. If a ray is rotated about its endpoint until it reaches the position , the angle produced is called (or simply…

3.2.2

Different Systems of measurement of angle

There are three standard systems for assigning a numerical measure to an angle.

3.2.3

Degree Measure

A degree, denoted , comes from splitting one complete rotation ( total) into equal parts, so is exactly of a full rotation.

3.2.4

Angles in Standard Position

An angle is in standard position when its vertex sits at the origin of a coordinate system and its initial side lies along the positive -axis.

3.2.5

Coterminal angles

Rotating the ray anticlockwise through one full turn produces an angle of but lands the terminal side back exactly where it started.

3.2.6

Basic Trigonometric ratios using a right triangle

For a right triangle with one acute angle , label the side opposite , the side adjacent to , and the hypotenuse.

3.2.7

Exact values of trigonometric functions of widely used angles

For a handful of frequently used angles, the trigonometric ratios work out to clean, exact values rather than messy decimals — worth memorising rather than recomputing each time.

3.2.8

Basic Trigonometric Identities

A trigonometric identity is an equation in trigonometric ratios that holds for every admissible value of the angle — not merely for some special value.

+Exercise 3.1i12 questions
  1. Q1Identify the quadrant in which an angle of each given measure lies: (i) $25^\circ$ (ii) $825^\circ$ (iii) $-55^\circ$ (iv) $328^\circ$ (v) $…Free
  2. Q2For each given angle, find a coterminal angle $\theta$ such that $0^\circ \le \theta < 360^\circ$: (i) $395^\circ$ (ii) $525^\circ$ (iii) $1…Free
  3. Q3If $a\cos\theta - b\sin\theta = c$, show that $a\sin\theta + b\cos\theta = \pm\sqrt{a^2+b^2-c^2}$.Free
  4. Q4If $\sin\theta+\cos\theta=m$, show that $\cos^6\theta+\sin^6\theta = \dfrac{4-3(m^2-1)^2}{4}$, where $m^2 \le 2$.Preview
  5. Q5If $\dfrac{\cos^4\alpha}{\cos^2\beta}+\dfrac{\sin^4\alpha}{\sin^2\beta}=1$, prove that (i) $\sin^4\alpha+\sin^4\beta=2\sin^2\alpha\sin^2\bet…Preview
  6. Q6If $y = \dfrac{2\sin\alpha}{1+\cos\alpha+\sin\alpha}$, then prove that $\dfrac{1-\cos\alpha+\sin\alpha}{1+\sin\alpha}=y$.Preview
  7. Q7If $x=\displaystyle\sum_{n=0}^{\infty}\cos^{2n}\theta$, $y=\displaystyle\sum_{n=0}^{\infty}\sin^{2n}\theta$ and $z=\displaystyle\sum_{n=0}^{…Preview
  8. Q8If $\tan^2\theta = 1-k^2$, show that $\sec\theta + \tan^3\theta\csc\theta = (2-k^2)^{3/2}$. Also, find the values of $k$ for which this resu…Preview
  9. Q9If $\sec\theta+\tan\theta=p$, obtain the values of $\sec\theta$, $\tan\theta$ and $\sin\theta$ in terms of $p$.Preview
  10. Q10If $\cot\theta(1+\sin\theta)=4m$ and $\cot\theta(1-\sin\theta)=4n$, then prove that $(m^2-n^2)^2=mn$.Preview
  11. Q11If $\csc\theta - \sin\theta = a^3$ and $\sec\theta - \cos\theta = b^3$, then prove that $a^2b^2(a^2+b^2)=1$.Preview
  12. Q12Eliminate $\theta$ from the equations $a\sec\theta - c\tan\theta = b$ and $b\sec\theta + d\tan\theta = c$.Preview
3.3

Radian Measure

Right-triangle trigonometry, built on degree measure, is genuinely limited: a right triangle can only ever contain acute angles, so this approach alone can never define or for an angle of, say, or .

3.3.1

Relationship between Degree and Radian Measures

Degrees and radians are two different units for measuring the same underlying quantity — an angle — much as Celsius and Fahrenheit are two units for temperature.

+Exercise 3.2i11 questions
  1. Q1Express each of the following angles in radian measure: (i) $30^\circ$ (ii) $135^\circ$ (iii) $-205^\circ$ (iv) $150^\circ$ (v) $330^\circ$Free
  2. Q2Find the degree measure corresponding to the following radian measures: (i) $\dfrac{\pi}{3}$ (ii) $\dfrac{\pi}{9}$ (iii) $\dfrac{2\pi}{5}$ (…Free
  3. Q3What must be the radius of a circular running path, around which an athlete must run $5$ times in order to describe $1$ km?Free
  4. Q4In a circle of diameter $40$ cm, a chord is of length $20$ cm. Find the length of the minor arc of the chord.Preview
  5. Q5Find the degree measure of the angle subtended at the centre of a circle of radius $100$ cm by an arc of length $22$ cm.Preview
  6. Q6What is the length of the arc intercepted by a central angle of measure $41^\circ$ in a circle of radius $10$ ft?Preview
  7. Q7If in two circles, arcs of the same length subtend angles $60^\circ$ and $75^\circ$ at the centre, find the ratio of their radii.Preview
  8. Q8The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius. Express the angle o…Preview
  9. Q9An airplane propeller rotates $1000$ times per minute. Find the number of degrees that a point on the edge of the propeller will rotate in $…Preview
  10. Q10A train is moving on a circular track of $1500$ m radius at the rate of $66$ km/hr. What angle will it turn in $20$ seconds?Preview
  11. Q11A circular metallic plate of radius $8$ cm and thickness $6$ mm is melted and molded into a pie (a sector of the circle with thickness) of r…Preview
3.4

Trigonometric functions and their properties

In earlier classes you met the trigonometric ratios of an angle only through a right triangle, which forces the angle to be acute — strictly between and .

3.4.1

Trigonometric Functions of any angle in terms of Cartesian coordinates

Setting up standard position. Place the vertex of an angle at the origin with its initial side along the positive -axis; the angle is measured (anticlockwise for positive , clockwise for negative ) to…

3.4.2

Trigonometric Functions of real numbers

Why extend to real numbers? Calculus, physics and chemistry frequently need and of a plain real number — not an angle measured in degrees — because these functions are used to model things like waves…

3.4.3

Allied Angles

Definition. Two angles are called allied angles if their sum or their difference is an integer multiple of (i.e. ). So, relative to a given angle , every angle of the form is allied to .

3.4.4

Some Characteristics of Trigonometric Functions

Trigonometric functions have two structural properties that make them exceptionally useful for modelling repeating phenomena: they repeat their values at regular intervals (periodicity), and they have…

+Exercise 3.3i6 questions
  1. Q1Find the values of (i) $\sin(480^\circ)$ (ii) $\sin(-1110^\circ)$ (iii) $\cos(300^\circ)$ (iv) $\tan(1050^\circ)$ (v) $\cot(660^\circ)$ (vi)…Free
  2. Q2$\left(\dfrac57,\ \dfrac{2\sqrt6}{7}\right)$ is a point on the terminal side of an angle $\theta$ in standard position. Determine the trigon…Free
  3. Q3Find the values of other five trigonometric functions for the following: (i) $\cos\theta=-\dfrac12$, $\theta$ lies in the III quadrant. (ii)…Preview
  4. Q4Prove that $\dfrac{\cot(180^\circ+\theta)\,\sin(90^\circ-\theta)\,\cos(-\theta)}{\sin(270^\circ+\theta)\,\tan(-\theta)\,\operatorname{cosec}…Preview
  5. Q5Find all the angles between $0^\circ$ and $360^\circ$ which satisfy the equation $\sin^2\theta=\dfrac34$.Preview
  6. Q6Show that $\sin^2\dfrac{\pi}{18}+\sin^2\dfrac{\pi}{9}+\sin^2\dfrac{7\pi}{18}+\sin^2\dfrac{4\pi}{9}=2.$Preview
3.5

Trigonometric Identities

A compound angle is any angle written as an algebraic sum or difference of two (or more) other angles, such as , , or .

3.5.1

Sum and difference identities or compound angles formulas

Identity 3.1 — .

+Exercise 3.4i25 questions
  1. Q1If $\sin x = \dfrac{15}{17}$ and $\cos y = \dfrac{12}{13}$, $0 < x < \dfrac{\pi}{2}$, $0 < y < \dfrac{\pi}{2}$, find the value of (i) $\sin(…Free
  2. Q2If $\sin A = \dfrac{3}{5}$ and $\cos B = \dfrac{9}{41}$, $0 < A < \dfrac{\pi}{2}$, $0 < B < \dfrac{\pi}{2}$, find the value of (i) $\sin(A+B…Free
  3. Q3Find $\cos(x-y)$, given that $\cos x = -\dfrac{4}{5}$ with $\pi < x < \dfrac{3\pi}{2}$ and $\sin y = -\dfrac{24}{25}$ with $\pi < y < \dfrac…Free
  4. Q4Find $\sin(x-y)$, given that $\sin x = \dfrac{8}{17}$ with $0 < x < \dfrac{\pi}{2}$ and $\cos y = -\dfrac{24}{25}$ with $\pi < y < \dfrac{3\…Preview
  5. Q5Find the value of (i) $\cos 105^\circ$ (ii) $\sin 105^\circ$ (iii) $\tan \dfrac{7\pi}{12}$.Preview
  6. Q6Prove that (i) $\cos(30^\circ + x) = \dfrac{\sqrt3 \cos x - \sin x}{2}$ (ii) $\cos(\pi + \theta) = -\cos\theta$ (iii) $\sin(\pi+\theta) = -\…Preview
  7. Q7Find a quadratic equation whose roots are $\sin 15^\circ$ and $\cos 15^\circ$.Preview
  8. Q8Expand $\cos(A+B+C)$. Hence prove that $$\cos A \cos B \cos C = \sin A \sin B \cos C + \sin B \sin C \cos A + \sin C \sin A \cos B,$$ if $A+…Preview
  9. Q9Prove that (i) $\sin(45^\circ+\theta) - \sin(45^\circ-\theta) = \sqrt2 \sin\theta$. (ii) $\sin(30^\circ+\theta) + \cos(60^\circ+\theta) = \c…Preview
  10. Q10If $a\cos(x+y) = b\cos(x-y)$, show that $(a+b)\tan x = (a-b)\cot y$.Preview
  11. Q11Prove that $\sin 105^\circ + \cos 105^\circ = \cos 45^\circ$.Preview
  12. Q12Prove that $\sin 75^\circ - \sin 15^\circ = \cos 105^\circ + \cos 15^\circ$.Preview
  13. Q13Show that $\tan 75^\circ + \cot 75^\circ = 4$.Preview
  14. Q14Prove that $\cos(A+B)\cos C - \cos(B+C)\cos A = \sin B \sin(C-A)$.Preview
  15. Q15Prove that $\sin(n+1)\theta \sin(n-1)\theta + \cos(n+1)\theta \cos(n-1)\theta = \cos 2\theta$, $n \in \mathbb{Z}$.Preview
  16. Q16If $x\cos\theta = y\cos\left(\theta + \dfrac{2\pi}{3}\right) = z\cos\left(\theta + \dfrac{4\pi}{3}\right)$, find the value of $xy+yz+zx$.Preview
  17. Q17Prove that (i) $\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$ (ii) $\cos(A+B)\cos(A-B) = \cos^2 A - \sin^2 B = \cos^2 B - \sin^2 A$ (iii) $\sin^…Preview
  18. Q18Show that $\cos^2 A + \cos^2 B - 2\cos A \cos B \cos(A+B) = \sin^2(A+B)$.Preview
  19. Q19If $\cos(\alpha-\beta) + \cos(\beta-\gamma) + \cos(\gamma-\alpha) = -\dfrac{3}{2}$, then prove that $$\cos\alpha + \cos\beta + \cos\gamma =…Preview
  20. Q20Show that (i) $\tan(45^\circ+A) = \dfrac{1+\tan A}{1-\tan A}$ (ii) $\tan(45^\circ-A) = \dfrac{1-\tan A}{1+\tan A}$.Preview
  21. Q21Prove that $\cot(A+B) = \dfrac{\cot A \cot B - 1}{\cot A + \cot B}$.Preview
  22. Q22If $\tan x = \dfrac{n}{n+1}$ and $\tan y = \dfrac{1}{2n+1}$, find $\tan(x+y)$.Preview
  23. Q23Prove that $\tan\left(\dfrac{\pi}{4}+\theta\right)\tan\left(\dfrac{3\pi}{4}+\theta\right) = -1$.Preview
  24. Q24Find the value of $\tan(\alpha+\beta)$, given that $\cot\alpha = \dfrac12$, $\alpha \in \left(\pi, \dfrac{3\pi}{2}\right)$ and $\sec\beta =…Preview
  25. Q25If $\theta+\phi = \alpha$ and $\tan\theta = k\tan\phi$, then prove that $\sin(\theta-\phi) = \dfrac{k-1}{k+1}\sin\alpha$.Preview
3.5.2

Multiple angle identities and submultiple angle identities

Multiple angles of are -- the angle scaled up by a whole number -- while sub-multiple angles are -- the angle scaled down.

+Exercise 3.5i11 questions
  1. Q1Find the value of $\cos 2A$, where $A$ lies in the first quadrant, when (i) $\cos A = \dfrac{15}{17}$ (ii) $\sin A = \dfrac{4}{5}$ (iii) $\t…Free
  2. Q2If $\theta$ is an acute angle, then find (i) $\sin\left(\dfrac{\pi}{4}-\dfrac{\theta}{2}\right)$, when $\sin\theta = \dfrac{1}{25}$. (ii) $\…Free
  3. Q3If $\cos\theta = \dfrac12\left(a+\dfrac1a\right)$, show that $\cos3\theta = \dfrac12\left(a^3+\dfrac1{a^3}\right)$.Free
  4. Q4Prove that $\cos5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta$.Preview
  5. Q5Prove that $\sin4\alpha = \dfrac{4\tan\alpha\left(1-\tan^2\alpha\right)}{\left(1+\tan^2\alpha\right)^2}$.Preview
  6. Q6If $A+B=45^\circ$, show that $(1+\tan A)(1+\tan B)=2$.Preview
  7. Q7Prove that $(1+\tan1^\circ)(1+\tan2^\circ)(1+\tan3^\circ)\cdots(1+\tan44^\circ)$ is a multiple of $4$.Preview
  8. Q8Prove that $\tan\left(\dfrac{\pi}{4}+\theta\right) - \tan\left(\dfrac{\pi}{4}-\theta\right) = 2\tan2\theta$.Preview
  9. Q9Show that $\cot\left(7\dfrac12\right)^\circ = \sqrt2+\sqrt3+\sqrt4+\sqrt6$.Preview
  10. Q10Prove that $(1+\sec2\theta)(1+\sec4\theta)\cdots(1+\sec2^n\theta) = \tan2^n\theta\,\cot\theta$.Preview
  11. Q11Prove that $32\sqrt3\,\sin\dfrac{\pi}{48}\cos\dfrac{\pi}{48}\cos\dfrac{\pi}{24}\cos\dfrac{\pi}{12}\cos\dfrac{\pi}{6} = 3$.Preview
3.5.3

Product to Sum and Sum to Product Identities

Some problems are far easier once a product of trigonometric functions is rewritten as a sum or difference (this is especially useful later when integrating a product of sines/cosines), and other prob…

+Exercise 3.6i14 questions
  1. Q1Express each of the following as a sum or difference: (i) $\sin 35^\circ \cos 28^\circ$ (ii) $\sin 4x \cos 2x$ (iii) $2\sin 10\theta \cos 2\…Free
  2. Q2Express each of the following as a product: (i) $\sin 75^\circ - \sin 35^\circ$ (ii) $\cos 65^\circ + \cos 15^\circ$ (iii) $\sin 50^\circ +…Free
  3. Q3Show that $\sin 12^\circ \sin 48^\circ \sin 54^\circ = \dfrac{1}{8}$.Free
  4. Q4Show that $\cos\dfrac{\pi}{15}\cos\dfrac{2\pi}{15}\cos\dfrac{3\pi}{15}\cos\dfrac{4\pi}{15}\cos\dfrac{5\pi}{15}\cos\dfrac{6\pi}{15}\cos\dfrac…Preview
  5. Q5Show that $\dfrac{\sin 8x \cos x - \sin 6x \cos 3x}{\cos 2x \cos x - \sin 3x \sin 4x} = \tan 2x$.Preview
  6. Q6Show that $\dfrac{(\cos\theta - \cos 3\theta)(\sin 8\theta + \sin 2\theta)}{(\sin 5\theta - \sin\theta)(\cos 4\theta - \cos 6\theta)} = 1$.Preview
  7. Q7Prove that $\sin x + \sin 2x + \sin 3x = \sin 2x\,(1 + 2\cos x)$.Preview
  8. Q8Prove that $\dfrac{\sin 4x + \sin 2x}{\cos 4x + \cos 2x} = \tan 3x$.Preview
  9. Q9Prove that $1 + \cos 2x + \cos 4x + \cos 6x = 4\cos x \cos 2x \cos 3x$.Preview
  10. Q10Prove that $\sin\dfrac{\theta}{2}\sin\dfrac{7\theta}{2} + \sin\dfrac{3\theta}{2}\sin\dfrac{11\theta}{2} = \sin 2\theta \sin 5\theta$.Preview
  11. Q11Prove that $\cos(30^\circ - A)\cos(30^\circ + A) + \cos(45^\circ - A)\cos(45^\circ + A) = \cos 2A + \dfrac{1}{4}$.Preview
  12. Q12Prove that $\dfrac{\sin x + \sin 3x + \sin 5x + \sin 7x}{\cos x + \cos 3x + \cos 5x + \cos 7x} = \tan 4x$.Preview
  13. Q13Prove that $\dfrac{\sin(4A - 2B) + \sin(4B - 2A)}{\cos(4A - 2B) + \cos(4B - 2A)} = \tan(A + B)$.Preview
  14. Q14Show that $\cot(A + 15^\circ) - \tan(A - 15^\circ) = \dfrac{4\cos 2A}{1 + 2\sin 2A}$.Preview
3.5.4

Conditional Trigonometric Identities

An identity in the usual sense (like ) is true for every admissible value of the angle, with no extra assumption needed.

+Exercise 3.7i5 questions
  1. Q1If $A + B + C = 180^\circ$, prove that (i) $\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C$ (ii) $\cos A + \cos B - \cos C = -1 + 4\cos…Free
  2. Q2If $A + B + C = 2s$, then prove that $\sin(s - A)\sin(s - B) + \sin s \sin(s - C) = \sin A \sin B$.Free
  3. Q3If $x + y + z = xyz$, then prove that $\dfrac{2x}{1 - x^2} + \dfrac{2y}{1 - y^2} + \dfrac{2z}{1 - z^2} = \dfrac{2x}{1 - x^2}\cdot\dfrac{2y}{…Preview
  4. Q4If $A + B + C = \dfrac{\pi}{2}$, prove the following (i) $\sin 2A + \sin 2B + \sin 2C = 4\cos A \cos B \cos C$ (ii) $\cos 2A + \cos 2B + \co…Preview
  5. Q5If $\triangle ABC$ is a right triangle and if $\angle A = \dfrac{\pi}{2}$, then prove that (i) $\cos^2 B + \cos^2 C = 1$ (ii) $\sin^2 B + \s…Preview
3.6

Trigonometric equations

A trigonometric equation is an equation in which the unknown appears only inside a trigonometric ratio — for instance or .

+Exercise 3.8i3 questions
  1. Q1Find the principal solution and general solutions of the following: (i) $\sin\theta = -\dfrac{1}{\sqrt2}$ (ii) $\cot\theta = \sqrt3$ (iii) $…Free
  2. Q2Solve the following equations for which solutions lies in the interval $0^\circ \le \theta < 360^\circ$: (i) $\sin^4 x = \sin^2 x$ (ii) $2\c…Preview
  3. Q3Solve the following equations: (i) $\sin 5x - \sin x = \cos 3x$ (ii) $2\cos^2\theta + 3\sin\theta - 3 = 0$ (iii) $\cos\theta + \cos 3\theta…Preview
3.7

Properties of Triangle

Every triangle has six basic elements — its three sides and three angles — and solving a triangle means finding all six once enough of them are known.

3.7.1

Law of Sines

When to use the Law of Sines. It is the right tool in exactly two situations: (i) to find an unknown angle, when two sides and a non-included angle are known (an SSA-type data set), and (ii) to find a…

3.7.2

Law of Cosines

When to use the Law of Cosines. The sine rule cannot solve a triangle from two sides and the included angle (SAS), or from all three sides (SSS) — those data sets need the Law of Cosines.

3.7.3

Projection Formula

Theorem 3.4 (Projection Formula). In ,

3.7.4

Area of the Triangle

The familiar formula needs a height, and for an oblique triangle no height is directly given — so the height is first expressed using the sine of an angle.

3.7.5

Half-Angle formula

Write for the semi-perimeter of . The half-angles of the triangle can be expressed purely in terms of the three sides and .

+Exercise 3.9i11 questions
  1. Q1In a $\triangle ABC$, if $\dfrac{\sin A}{\sin C}=\dfrac{\sin(A-B)}{\sin(B-C)}$, prove that $a^2, b^2, c^2$ are in Arithmetic Progression.Free
  2. Q2The angles of a triangle $ABC$ are in Arithmetic Progression and if $b:c=\sqrt3:\sqrt2$, find $\angle A$.Free
  3. Q3In a $\triangle ABC$, if $\cos C=\dfrac{\sin A}{2\sin B}$, show that the triangle is isosceles.Free
  4. Q4In a $\triangle ABC$, prove that $\dfrac{\sin B}{\sin C}=\dfrac{c-a\cos B}{b-a\cos C}$.Preview
  5. Q5In a $\triangle ABC$, prove that $a\cos A+b\cos B+c\cos C=2a\sin B\sin C$.Preview
  6. Q6In a $\triangle ABC$, $\angle A=60^\circ$. Prove that $b+c=2a\cos\left(\dfrac{B-C}{2}\right)$.Preview
  7. Q7In a $\triangle ABC$, prove the following (i) $a\sin\left(\dfrac{A}{2}+B\right)=(b+c)\sin\dfrac{A}{2}$ (ii) $a(\cos B+\cos C)=2(b+c)\sin^2\d…Preview
  8. Q8In a $\triangle ABC$, prove that $(a^2-b^2+c^2)\tan B=(a^2+b^2-c^2)\tan C$.Preview
  9. Q9An Engineer has to develop a triangular shaped park with a perimeter $120$ m in a village. The park to be developed must be of maximum area.…Preview
  10. Q10A rope of length $12$ m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so forme…Preview
  11. Q11Derive the Projection formula from (i) Law of Sines, (ii) Law of Cosines.Preview
3.8

Application to Triangle

Solving a triangle means computing every one of its six elements -- the three sides and the three angles -- once enough of them are already known.

+Exercise 3.10i16 questions
  1. Q1Determine whether the following measurements produce one triangle, two triangles, or no triangle: $\angle B = 88^\circ$, $a = 23$, $b = 2$.…Free
  2. Q2If the sides of a $\triangle ABC$ are $a = 4$, $b = 6$ and $c = 8$, then show that $4\cos B + 3\cos C = 2$.Free
  3. Q3In a $\triangle ABC$, if $a = \sqrt3 - 1$, $b = \sqrt3 + 1$ and $C = 60^\circ$, find the other side and the other two angles.Free
  4. Q4In any $\triangle ABC$, prove that the area $\triangle = \dfrac{b^2+c^2-a^2}{4\cot A}$.Preview
  5. Q5In a $\triangle ABC$, if $a = 12$ cm, $b = 8$ cm and $C = 30^\circ$, then show that its area is $24$ sq.cm.Preview
  6. Q6In a $\triangle ABC$, if $a = 18$ cm, $b = 24$ cm and $c = 30$ cm, then show that its area is $216$ sq.cm.Preview
  7. Q7Two soldiers $A$ and $B$ in two different underground bunkers on a straight road spot an intruder at the top of a hill. The angle of elevati…Preview
  8. Q8A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point $P$, he find…Preview
  9. Q9Two Navy helicopters $A$ and $B$ are flying over the Bay of Bengal at the same altitude from the sea level to search for a missing boat. Pil…Preview
  10. Q10A straight tunnel is to be made through a mountain. A surveyor observes the two extremities $A$ and $B$ of the tunnel to be built from a poi…Preview
  11. Q11A farmer wants to purchase a triangular shaped land with sides $120$ feet and $60$ feet and the angle included between these two sides is $6…Preview
  12. Q12A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depress…Preview
  13. Q13A plane is $1$ km from one landmark and $2$ km from another. From the plane's point of view, the land between them subtends an angle of $45^…Preview
  14. Q14A man starts his morning walk at a point $A$, reaches two points $B$ and $C$ and finally comes back to $A$ such that $\angle A = 60^\circ$ a…Preview
  15. Q15Two vehicles leave the same place $P$ at the same time moving along two different roads. One vehicle moves at an average speed of $60$ km/hr…Preview
  16. Q16Suppose that a satellite in space, an earth station and the centre of the earth all lie in the same plane. Let $r$ be the radius of the eart…Preview
3.9

Inverse Trigonometric Functions

22 Q

A function has an inverse only when it is one-to-one (injective) and onto (surjective) on the domain in use — an inverse simply cannot be defined for a function that is not one-to-one, because then a…

+Exercise 3.11i2 questions
  1. Q1Find the principal value of (i) $\sin^{-1}\dfrac1{\sqrt2}$ (ii) $\cos^{-1}\dfrac{\sqrt3}2$ (iii) $\csc^{-1}(-1)$ (iv) $\sec^{-1}(-\sqrt2)$ (…Free
  2. Q2A man standing directly opposite to one side of a road of width $x$ metre views a circular shaped traffic green signal of diameter $a$ metre…Preview
+Exercise 3.12i20 questions
  1. Q1$\dfrac1{\cos80^\circ}-\dfrac{\sqrt3}{\sin80^\circ}=$ (1) $\sqrt2$ (2) $\sqrt3$ (3) $2$ (4) $4$Free
  2. Q2If $\cos28^\circ+\sin28^\circ=k^3$, then $\cos17^\circ$ is equal to (1) $\dfrac{k^3}{\sqrt2}$ (2) $-\dfrac{k^3}{\sqrt2}$ (3) $\pm\dfrac{k^3}…Free
  3. Q3The maximum value of $4\sin^2x+3\cos^2x+\sin\dfrac x2+\cos\dfrac x2$ is (1) $4+\sqrt2$ (2) $3+\sqrt2$ (3) $9$ (4) $4$Free
  4. Q4$\left(1+\cos\dfrac\pi8\right)\left(1+\cos\dfrac{3\pi}8\right)\left(1+\cos\dfrac{5\pi}8\right)\left(1+\cos\dfrac{7\pi}8\right)=$ (1) $\dfrac…Preview
  5. Q5If $\pi<2\theta<\dfrac{3\pi}2$, then $\sqrt{2+\sqrt{2+2\cos4\theta}}$ equals (1) $-2\cos\theta$ (2) $-2\sin\theta$ (3) $2\cos\theta$ (4) $2\…Preview
  6. Q6If $\tan40^\circ=\lambda$, then $\dfrac{\tan140^\circ-\tan130^\circ}{1+\tan140^\circ\tan130^\circ}=$ (1) $\dfrac{1-\lambda^2}\lambda$ (2) $\…Preview
  7. Q7$\cos1^\circ+\cos2^\circ+\cos3^\circ+\cdots+\cos179^\circ=$ (1) $0$ (2) $1$ (3) $-1$ (4) $89$Preview
  8. Q8Let $f_k(x)=\dfrac1k\left(\sin^kx+\cos^kx\right)$ where $x\in\mathbb R$ and $k\ge1$. Then $f_4(x)-f_6(x)=$ (1) $\dfrac14$ (2) $\dfrac1{12}$…Preview
  9. Q9Which of the following is not true? (1) $\sin\theta=-\dfrac34$ (2) $\cos\theta=-1$ (3) $\tan\theta=25$ (4) $\sec\theta=\dfrac14$Preview
  10. Q10$\cos2\theta\cos2\phi+\sin^2(\theta-\phi)-\sin^2(\theta+\phi)$ is equal to (1) $\sin2(\theta+\phi)$ (2) $\cos2(\theta+\phi)$ (3) $\sin2(\the…Preview
  11. Q11$\dfrac{\sin(A-B)}{\cos A\cos B}+\dfrac{\sin(B-C)}{\cos B\cos C}+\dfrac{\sin(C-A)}{\cos C\cos A}$ is (1) $\sin A+\sin B+\sin C$ (2) $1$ (3)…Preview
  12. Q12If $\cos p\theta+\cos q\theta=0$ and if $p\ne q$, then $\theta$ is equal to ($n$ is any integer) (1) $\dfrac{\pi(3n+1)}{p-q}$ (2) $\dfrac{\p…Preview
  13. Q13If $\tan\alpha$ and $\tan\beta$ are the roots of $x^2+ax+b=0$, then $\dfrac{\sin(\alpha+\beta)}{\sin\alpha\sin\beta}$ is equal to (1) $\dfra…Preview
  14. Q14In a triangle $ABC$, $\sin^2A+\sin^2B+\sin^2C=2$, then the triangle is (1) an equilateral triangle (2) an isosceles triangle (3) a right tri…Preview
  15. Q15If $f(\theta)=|\sin\theta|+|\cos\theta|,\ \theta\in\mathbb R$, then $f(\theta)$ is in the interval (1) $[0,2]$ (2) $\left[1,\sqrt2\right]$ (…Preview
  16. Q16$\dfrac{\cos6x+6\cos4x+15\cos2x+10}{\cos5x+5\cos3x+10\cos x}$ is equal to (1) $\cos2x$ (2) $\cos x$ (3) $\cos3x$ (4) $2\cos x$Preview
  17. Q17The triangle of maximum area with constant perimeter $12\,\text m$ (1) is an equilateral triangle with side $4\,\text m$ (2) is an isosceles…Preview
  18. Q18A wheel is spinning at $2$ radians/second. How many seconds will it take to make $10$ complete rotations? (1) $10\pi$ seconds (2) $20\pi$ se…Preview
  19. Q19If $\sin\alpha+\cos\alpha=b$, then $\sin2\alpha$ is equal to (1) $b^2-1$, if $b\le\sqrt2$ (2) $b^2-1$, if $b>\sqrt2$ (3) $b^2-1$, if $b\ge1$…Preview
  20. Q20In a $\triangle ABC$, if (i) $\sin\dfrac A2\sin\dfrac B2\sin\dfrac C2>0$ (ii) $\sin A\sin B\sin C>0$ then (1) Both (i) and (ii) are true (2)…Preview
3.10

Summary

This chapter's results fall into a small number of formula families; gathering them here is a working reference, not a re-teach.

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 39 questions39 questions
  1. Q1With usual notations, area of the triangle ABC is: (a) $\frac{1}{2}ab\cos A$ (b) $\frac{1}{2}ab\cos C$ (c) $\frac{1}{2}bc\sin B$ (d) $\frac{…Preview
  2. Q2The principal value of $\text{cosec}^{-1}(-2)$ is: (a) $\dfrac{-\pi}{3}$ (b) $\dfrac{-\pi}{6}$ (c) $\dfrac{\pi}{6}$ (d) $\dfrac{\pi}{3}$Preview
  3. Q3If $\tan\theta + \sin\theta = p$, $\tan\theta - \sin\theta = q$ and $p > q$ then show that $p^2 - q^2 = 4\sqrt{pq}$.Preview
  4. Q4(a) State and prove any one of the Napier's formulae. **OR** (b) If $g(x) = x^2+2x+1$ and $g[f(x)] = 4x^2-12x+9$ then find the values of $f(…Preview
  5. Q5Which of the following is not a periodic function with period $2\pi$? (a) $\tan x$ (b) $\cos x$ (c) $\sin x$ (d) $\text{cosec}\,x$Preview
  6. Q6The minimum and the maximum values of $|\cos x|-2$ are respectively: (a) 0 and 2 (b) $-2$ and 0 (c) $-2$ and $-1$ (d) $-1$ and 1Preview
  7. Q7A football player can kick a football from ground level with an initial velocity (u) of 80 ft/second. Find the maximum horizontal distance t…Preview
  8. Q8(a) For the given base curve $y=\sin x$, draw $y=\dfrac{1}{2}\sin 2x$. **OR** (b) Write any five different forms of an equation of a straigh…Preview
  9. Q9(a) State and prove any one of the Napier's formulae. **OR** (b) Do the limit of the function $\dfrac{\sin(x-\lfloor x\rfloor)}{x-\lfloor x\…Preview
  10. Q10In a triangle ABC, $\sin^2 A + \sin^2 B + \sin^2 C = 2$, then the triangle is ________. (a) Equilateral triangle (b) Isosceles triangle (c)…Preview
  11. Q11Find the principal solution of $\cos\theta = \dfrac{-1}{2}$.Preview
  12. Q12Find the domain of $\dfrac{1}{1-2\sin x}$.Preview
  13. Q13(a) If $A+B+C=\pi$, prove that $\cos^2 A + \cos^2 B + \cos^2 C = 1 - 2\cos A \cos B \cos C$. **OR** (b) If $\dfrac{\log x}{y-z} = \dfrac{\lo…Preview
  14. Q14The value of $\tan 90°$ is: (a) $\frac{\sqrt{3}}{2}$ (b) 0 (c) 1 (d) $\infty$Preview
  15. Q15$\sec(-\theta)$ is: (a) $\sec\theta$ (b) $\cos\theta$ (c) $\sec(-\theta)$ (d) $\cos(-\theta)$Preview
  16. Q16Express $\sin 50° + \sin 20°$ as a product.Preview
  17. Q17Find the value of $\cos 135°$.Preview
  18. Q18(a) Prove that: $\dfrac{\cot(180°+\theta)\, \sin(90°-\theta)\, \cos(-\theta)}{\sin(270°+\theta)\, \tan(-\theta)\, \text{cosec}(360°+\theta)}…Preview
  19. Q19Which of the following is not true? (a) $\tan\theta = 25$ (b) $\sin\theta = -\dfrac{3}{4}$ (c) $\sec\theta = \dfrac{1}{4}$ (d) $\cos\theta =…Preview
  20. Q20$\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \ldots + \cos 179^\circ =$ (a) $-1$ (b) 0 (c) 89 (d) 1Preview
  21. Q21If $A + B = 45^\circ$, show that $(1+\tan A)(1+\tan B) = 2$.Preview
  22. Q22Find the range of the function $\dfrac{1}{2\cos x - 1}$.Preview
  23. Q23Prove that: $\dfrac{\sin 4x + \sin 2x}{\cos 4x + \cos 2x} = \tan 3x$Preview
  24. Q24If $A+B+C=180^\circ$, prove that $\tan\dfrac{A}{2}\tan\dfrac{B}{2} + \tan\dfrac{B}{2}\tan\dfrac{C}{2} + \tan\dfrac{C}{2}\tan\dfrac{A}{2} = 1…Preview
  25. Q25$\dfrac{\sin(A-B)}{\cos A\cos B}+\dfrac{\sin(B-C)}{\cos B\cos C}+\dfrac{\sin(C-A)}{\cos C\cos A}$ is: (a) $0$ (b) $\sin A+\sin B+\sin C$ (c)…Preview
  26. Q26Show that $\tan(45^\circ-A)=\dfrac{1-\tan A}{1+\tan A}$Preview
  27. Q27Find the value of $\cos 105^\circ$.Preview
  28. Q28Prove that: $\dfrac{\cos11^\circ+\sin11^\circ}{\cos11^\circ-\sin11^\circ}=\tan56^\circ$Preview
  29. Q29(a) State and prove Napier's Formula. **OR** (b) If the equation $\lambda x^2-10xy+12y^2+5x-16y-3=0$ represents a pair of straight lines, fi…Preview
  30. Q30In a triangle ABC, $\sin^2 A + \sin^2 B + \sin^2 C = 2$, then the triangle is: (a) right triangle (b) equilateral triangle (c) scalene trian…Preview
  31. Q31If $\cos 28^\circ + \sin 28^\circ = k^3$, then $\cos 17^\circ$ is equal to: (a) $\pm\dfrac{k^3}{\sqrt{2}}$ (b) $\dfrac{k^3}{\sqrt{2}}$ (c) $…Preview
  32. Q32Find the range of the function $f(x) = \dfrac{1}{1 - 3\cos x}$Preview
  33. Q33Prove that $\sin(45^\circ+\theta) - \sin(45^\circ-\theta) = \sqrt{2}\sin\theta$Preview
  34. Q34(a) Prove that $\dfrac{\cot(180^\circ+\theta)\sin(90^\circ-\theta)\cos(-\theta)}{\sin(270^\circ+\theta)\tan(-\theta)\text{cosec}(360^\circ+\…Preview
  35. Q35(a) Show that $\tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ = 3$ **OR** (b) Integrate : $\dfrac{1}{\sqrt{x^2+5x+4}}$Preview
  36. Q36$\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \ldots + \cos 179^\circ =$ (a) $-1$ (b) $0$ (c) $89$ (d) $1$Preview
  37. Q37Which of the following is not true? (a) $\tan\theta = 25$ (b) $\sin\theta = -\dfrac{3}{4}$ (c) $\sec\theta = \dfrac{1}{4}$ (d) $\cos\theta =…Preview
  38. Q38Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring $15^\circ$.Preview
  39. Q39If $\theta+\phi=\alpha$ and $\tan\theta=k\tan\phi$, then prove that $\sin(\theta-\phi)=\dfrac{k-1}{k+1}\sin\alpha$ **OR** Prove that $\sqrt[…Preview