Trigonometric Ration of Any Angle

6.2

Trigonometric Ration of Any Angle

The image illustrates the concept of trigonometric ratios. It features a right triangle labeled with sides: Hypotenuse, Opposite side, and Adjacent side. The triangle is used to explain the sine, cosine, and tangent functions. Below the triangle, there are three boxes: 1. The first box defines sine (sin θ) as the ratio of the opposite side to the hypotenuse (BC/AB). 2. The second box defines cosine (cos θ) as the ratio of the adjacent side to the hypotenuse (AC/AB). 3. The third box defines tangent (tan θ) as the ratio of the opposite side to the adjacent side (BC/AC).

Formula of Cosecant, Secant and Cotangent

Cosecant

\(\text{cosec }\theta= \dfrac{1}{\text{sin }\theta}\)

Secant

\(\text{sec }\theta= \dfrac{1}{\text{cos }\theta}\)

Cotangent

\(\text{cot }\theta= \dfrac{1}{\text{tan }\theta}\)

Complementary Angles

The angles \(A\) and \(B\) are complementary angles to each other if \(A+B=90^\circ\). Hence,

\(A=90^\circ-B\) and \(B=90^\circ-A\)

Formula of the Complementary Angles

\(\text{sin }\theta= \text{cos } (90^{\circ}-\theta)\)
\(\text{cos }\theta= \text{sin } (90^{\circ}-\theta)\)
\(\text{tan }\theta= \text{cot } (90^{\circ}-\theta)\)
\(\text{sec }\theta= \text{cosec } (90^{\circ}-\theta)\)
\(\text{cosec }\theta= \text{sec } (90^{\circ}-\theta)\)
\(\text{cot }\theta= \text{tan } (90^{\circ}-\theta)\)

\(4\) Methods to Determine the Values of the Trigonometric Ratios 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 of that particular angle.

Method \(2\): Use a Unit Circle

A visual depiction of a unit circle displaying four varied numbers positioned along its edge, emphasizing mathematical principles.

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 \( 0° \leqslant \theta \leqslant 2\pi\).

Four distinct graphs illustrating the same trigonometric ratio across all four quadrants for clear visual comparison.

Method \(4\): Use a Right-angled Triangle

The trigonometric ratios of special angles \(30^\circ\), \(45^\circ\) and \(60^\circ\) can be determined by using right-angled triangles.

Angle / Ratio		sin	cos	tan	cosec	sec	cot
\(30^\circ\)	\(\dfrac{\pi}{6}\)	\(\dfrac{1}{2}\)	\(\dfrac{\sqrt{3}}{2}\)	\(\dfrac{1}{\sqrt{3}}\)	\(2\)	\(\dfrac{2}{\sqrt{3}}\)	\(\sqrt{3}\)
\(45^\circ\)	\(\dfrac{\pi}{4}\)	\(\dfrac{1}{\sqrt{2}}\)	\(\dfrac{1}{\sqrt{2}}\)	\(1\)	\(\sqrt{2}\)	\(\sqrt{2}\)	\(1\)
\(60^\circ\)	\(\dfrac{\pi}{3}\)	\(\dfrac{\sqrt{3}}{2}\)	\(\dfrac{1}{2}\)	\(\sqrt{3}\)	\(\dfrac{2}{\sqrt{3}}\)	\(2\)	\(\dfrac{1}{\sqrt{3}}\)

Example \(1\)

Question

(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 \(k \gt 0\).
	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)

\(\begin{aligned} \text{cosec }\theta&= \text{sec } (90^{\circ}-\theta)\\ \text{cosec }27^{\circ}&= \text{sec } (90^{\circ}-27^{\circ})\\ &= \text{sec }63^{\circ}\\ &= \dfrac{1}{\text{cos }63^{\circ}}\\ &= \dfrac{1}{k}. \end{aligned}\)

Example \(2\)

Question

Use the unit circle 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, \(\text{cos }135^{\circ} = -\dfrac{1}{\sqrt2}\).