Mensuration and trigonometry Study Guide

Trigonometry provides powerful tools to find unknown side lengths or angles in right-angled triangles. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios relate the angles of a right-angled triangle to the lengths of its sides. For a given angle θ in a right-angled triangle, the side opposite the angle is called the 'opposite' side, the side adjacent to the angle (that is not the hypotenuse) is called the 'adjacent' side, and the longest side (opposite the right angle) is the 'hypotenuse'.

The definitions are:

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent

A common mnemonic to remember these is SOH CAH TOA.

When solving problems, first, clearly label the sides of the triangle relative to the known or unknown angle. Then, choose the appropriate trigonometric ratio based on the sides involved. For example, if you know the opposite side and the hypotenuse, and need to find the angle, you would use sin(θ). If you know an angle and the adjacent side, and need to find the hypotenuse, you would use cos(θ).

To find a side length, rearrange the chosen formula. To find an angle, use the inverse trigonometric functions: arcsin (sin⁻¹), arccos (cos⁻¹), or arctan (tan⁻¹). Always ensure your calculator is in degree mode for these calculations unless otherwise specified.

Exact trigonometric ratios refer to the precise values of sine, cosine, and tangent for specific angles, commonly 0°, 30°, 45°, 60°, and 90°. Unlike calculator approximations, these exact values are often expressed using surds (square roots) and fractions, providing a more accurate representation. Understanding these exact values is crucial for various mathematical problems where precision is paramount, and they form the basis for further study in trigonometry. These values are derived from properties of special right-angled triangles: an isosceles right-angled triangle (with angles 45°, 45°, 90°) and half of an equilateral triangle (resulting in angles 30°, 60°, 90°). For example, in a 45-45-90 triangle with hypotenuse length $\sqrt{2}$ and equal sides of length 1, sin(45°) = 1/$\sqrt{2}$ = $\sqrt{2}/2$. In a 30-60-90 triangle with hypotenuse 2, opposite side to 30° being 1, and opposite side to 60° being $\sqrt{3}$, we find sin(30°) = 1/2, cos(30°) = $\sqrt{3}/2$, and tan(30°) = 1/$\sqrt{3}$ = $\sqrt{3}/3$. Similarly, the values for 60° can be found, and understanding these exact values allows for solving problems without a calculator.