Trigonometric ratio of any angle

Trigonometric Ration of Any Angle

\(\text{sin }\theta= \text{cos } (90^{\circ}-\theta)\)

\(\text{cos }\theta= \text{sin } (90^{\circ}-\theta)\)

\(\text{tan }\theta= \text{cot } (90^{\circ}-\theta)\)

\(\text{sec }\theta= \text{cosec } (90^{\circ}-\theta)\)

\(\text{cosec }\theta= \text{sec } (90^{\circ}-\theta)\)

\(\text{cot }\theta= \text{tan } (90^{\circ}-\theta)\)

6.2

Trigonometric Ration of Any Angle


	\(\text{sin} \ \theta = \dfrac{\text{opposite side}}{\text{hypotenuse}} = \dfrac{BC}{AB}\)

	\(\text{kos} \ \theta = \dfrac{\text{adjacent side}}{\text{hypotenuse}} = \dfrac{AC}{AB}\)

	\(\text{tan} \ \theta = \dfrac{\text{opposite side}}{\text{adjacent side}} = \dfrac{BC}{AC}\)

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,

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