## Trigonometric Ration of Any Angle

 6.2 Trigonometric Ration of Any Angle

• The diagram below shows the triangle $$ABC$$

• The trigonometric ratios can be defined as follows:
 $$\text{sin} \ \theta = \dfrac{\text{opposite side}}{\text{hypotenuse}} = \dfrac{BC}{AB}$$ $$\text{kos} \ \theta = \dfrac{\text{adjacent side}}{\text{hypotenuse}} = \dfrac{AC}{AB}$$ $$\text{tan} \ \theta = \dfrac{\text{opposite side}}{\text{adjacent side}} = \dfrac{BC}{AC}$$

• There are three more ratios that are the reciprocals of these trigonometric ratios:
Cosecant
 $$\text{cosec }\theta= \dfrac{1}{\text{sin }\theta}$$
Secant
 $$\text{sec }\theta= \dfrac{1}{\text{cos }\theta}$$
Cotangent
 $$\text{cot }\theta= \dfrac{1}{\text{tan }\theta}$$

• The angles $$A$$ and $$B$$ are complementary angles to each other if $$A + B = 90^{\circ}$$. Hence,

 $$A=90^{\circ} -B \text{ and }B=90^{\circ}-A$$

• The formulae of the complementary angles are as follows:
 $$\text{sin }\theta= \text{cos } (90^{\circ}-\theta)$$ $$\text{cos }\theta= \text{sin } (90^{\circ}-\theta)$$ $$\text{tan }\theta= \text{cot } (90^{\circ}-\theta)$$ $$\text{sec }\theta= \text{cosec } (90^{\circ}-\theta)$$ $$\text{cosec }\theta= \text{sec } (90^{\circ}-\theta)$$ $$\text{cot }\theta= \text{tan } (90^{\circ}-\theta)$$

Example

 (a) Given that $$\text{sin } 77^{\circ} = 0.9744$$ and $$\text{cos } 77^{\circ} = 0.225$$. Find the value of $$\text{cos }13^{\circ}$$. (b) Given $$\text{cos }63^{\circ} = k$$, where $$k \text{ > }0$$. Find the value of $$\text{cosec }27^{\circ}$$ in terms of $$k$$. Solution: (a) \begin{aligned} \text{cos }\theta&= \text{sin } (90^{\circ}-\theta)\\ \text{cos }13^{\circ}&= \text{sin } (90^{\circ}-13^{\circ})\\ &= \text{sin }77^{\circ}\\ &= 0.9744 \end{aligned} (b) \begin{aligned} \text{cosec }\theta&= \text{sec } (90^{\circ}-\theta)\\ \text{cosec }27^{\circ}&= \text{sec } (90^{\circ}-27^{\circ})\\ &= \text{sec }63^{\circ}\\ &= \dfrac{1}{\text{cos }63^{\circ}}\\ &= \dfrac{1}{k} \end{aligned}

• $$4$$ methods to determine the values of the trigonometric rations for any angle:

Method 1: Use a calculator

• The values of sine, cosine and tangent can be determined by using a calculator

• The values for cosecant, secant and cotangent can be calculated by inversing the values of the trigonometric ratios of sine, cosine and tangent

Method 2: Use a unit circle

 Example Use the unit circle above and state the value of $$\text{cos }135^{\circ}$$. Solution: The coordinates that correspond to $$135^{\circ}$$ are $$\begin{pmatrix} -\dfrac{1}{\sqrt2}, \ \dfrac{1}{\sqrt2} \end{pmatrix}$$ and $$\text{cos }135^{\circ} = x\text{-coordinate}$$. Hence, $$\text{cos }135^{\circ} = -\dfrac{1}{\sqrt2}$$.

Method 3: Use the corresponding trigonometric ratio of the reference angle

• The diagram shows the reference angles, $$\alpha$$ for the angles $$0° \leqslant \theta \leqslant 360^{\circ}$$ or $$0° \leqslant \theta \leqslant 2\pi$$

Method 4: Use a right-angled triangle

• The trigonometric ratios of special angles $$30^{\circ}, 45^{\circ} \text{ and }60^{\circ}$$ can be determined by using right-angled triangles

 Angle \ Ratio sin cos tan cosec sec cot $$30^{\circ}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{1}{2}$$ $$\dfrac{\sqrt3}{2}$$ $$\dfrac{1}{\sqrt3}$$ $$2$$ $$\dfrac{2}{\sqrt3}$$ $$\sqrt3$$ $$45^{\circ}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{1}{\sqrt2}$$ $$\dfrac{1}{\sqrt2}$$ $$1$$ $$\sqrt2$$ $$\sqrt2$$ $$1$$ $$60^{\circ}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\sqrt3}{2}$$ $$\dfrac{1}{2}$$ $$\sqrt3$$ $$\dfrac{2}{\sqrt3}$$ $$2$$ $$\dfrac{1}{\sqrt3}$$