Ch 3Trigonometric Functions — Class 11 Mathematics, concept-first.

An angle is the amount of rotation of a ray about its fixed endpoint, carrying it from an initial position to a terminal position.

There are two systems commonly used to measure the size of an angle.

The definition of and as ratios of sides in a right triangle only makes sense for an acute angle (strictly between and ), since a right triangle cannot have an angle of or .

Theorem (the fundamental identity). For every real number ,

Signs in each quadrant. Since is the x-coordinate and the y-coordinate of the point on the unit circle (Section 3), the sign of each function is exactly the sign of or in that quadrant, and the sign o…

Theorem. For all real ,

The tangent and cotangent sum/difference formulas are not proved from the unit circle again; they are deduced algebraically from the sine and cosine formulas already established in Section 6, by divid…

The four sum-to-product formulas rewrite a sum or difference of two sine or cosine terms as a product -- the reverse operation to the sum/difference formulas of Section 6, and essential for simplifyin…

Double-angle identities. Setting in the Section 6 sum formulas gives, immediately, Using the fundamental identity to eliminate one squared term at a time from gives two further, equally useful, forms:…

Because every trigonometric function is periodic (Section 5), an equation such as is never satisfied by a single value of alone -- once one solution is known, so is every coterminal angle, and (for si…

Angles. An angle is a signed rotation (anticlockwise positive, clockwise negative) of a ray about its endpoint; two angles with the same terminal side, differing by a whole number of revolutions, are…