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Business Mathematics and Basic Statistics · Ch 6 — Trigonometry

Trigonometric Ratios

1

Trigonometric Ratios

Trigonometry is the branch of mathematics that studies the relationship between the angles and the sides of a triangle. In the WBCHSE Class 11 Commerce Business Mathematics and Basic Statistics syllabus, trigonometry is a compact, self-contained unit that builds the angle-ratio toolkit used later whenever an angle-based numerical situation appears.

Consider a right-angled triangle with one acute angle marked θ\theta. Relative to θ\theta, the three sides are named the opposite side (the side across from θ\theta), the adjacent side (the side next to θ\theta, not the hypotenuse), and the hypotenuse (the side opposite the right angle — always the longest side).

The six trigonometric ratios of θ\theta are defined as:

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

cosec θ=hypotenuseopposite,sec⁡θ=hypotenuseadjacent,cot⁡θ=adjacentopposite\text{cosec}\,\theta = \frac{\text{hypotenuse}}{\text{opposite}}, \qquad \sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}}, \qquad \cot\theta = \frac{\text{adjacent}}{\text{opposite}}

Note

Reciprocal Relationships

cosec θ=1sin⁡θ\text{cosec}\,\theta = \dfrac{1}{\sin\theta}, sec⁡θ=1cos⁡θ\sec\theta = \dfrac{1}{\cos\theta}, cot⁡θ=1tan⁡θ\cot\theta = \dfrac{1}{\tan\theta} — each ratio in the second row is the reciprocal of the matching ratio in the first row. It follows directly that tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta} and cot⁡θ=cos⁡θsin⁡θ\cot\theta = \dfrac{\cos\theta}{\sin\theta}.

The value of every trigonometric ratio depends only on the angle θ\theta, never on the size of the triangle drawn — two right triangles with the same acute angle θ\theta but different side lengths are similar triangles, so the ratio of any two corresponding sides is the same in both. This is exactly what makes it possible to build one fixed table of ratio values for a handful of standard angles and reuse it in any problem, which is the subject of the next section.

← Previous chapterCh 5 — Theory of Sets — IntroductionNext · 2 →Ratios of Standard Angles (0°, 30°, 45°, 60°, 90°)