# 10th Class Mathematics Introduction to Trigonometry

Introduction to Trigonometry

Category : 10th Class

Introduction to Trigonometry

• Trigonometry: The branch of mathematics that deals with the study of relationships between the sides and angles of a triangle is called trigonometry.

• The word 'trigonometry' is derived from the greek words 'tri' meaning three, 'gon' meaning sides and 'metron' meaning measure.

• Trigonometric ratios: In right AABC, AB is the hypotenuse, AB is the side opposite to $\angle C(=\theta )$, and BC is the side adjacent to 6.

The trigonometric ratios for angle $\theta$:

(a) $\sin \theta =\frac{side\,opposite\,to\,\theta }{Hypotenuse}=\frac{AB}{AC}$

(b) $\cos \theta =\frac{side\,adjacent\,to\,\theta }{Hypotenuse}=\frac{BC}{AC}$

(c) $\tan \theta =\frac{side\,opposite\,to\,\theta }{side\,adjacent\,to\,\theta }=\frac{AB}{BC}$

(d) $\cos ec\theta =\frac{Hypotenuse}{side\,opposite\,to\,\theta }=\frac{AC}{AB}$

(e) $sec\theta =\frac{Hypotenuse}{side\,adjacent\,to\,\theta }=\frac{AC}{BC}$

(f) $\cot \theta =\frac{side\,adjacent\,to\,\theta }{side\,opposite\,to\,\theta }=\frac{BC}{AB}$

• Full names of the trigonometric ratios: sin = sine; cos = cosine; tan = tangent; cosec = cosecant; sec = secant; cot = cotangent

Note: (a) $\sin \theta$ is a single symbol. It does not mean the product of sine and$\theta$.

(b) In short, t-ratios is used for trigonometric ratios.

(c) Trigonometric ratios are real numbers.

• Reciprocal relations: The reciprocal relations for the trigonometric ratios are
(a) $\frac{1}{\sin \theta }=\cos ec\theta$ (b) $\frac{1}{\cos \theta }=sec\theta$

(c) $\frac{1}{\tan \theta }=\cot \theta$

• Quotient relations:

(a) $\tan \theta =\frac{\sin \theta }{\cos \theta }$        (b) $\cot \theta =\frac{\cos \theta }{\sin \theta }$

• Values of trigonometric ratios: The values of trigonometric ratios remain the same for the same angle of different right triangles.

Note: A trigonometric ratio depends on the magnitude of the angle and not on its size.

• Trigonometric ratios of complementary angles:

If$\theta$ is acute, then

(a) $\sin ({{90}^{o}}-\theta )=\cos \theta$

(b) $\cos ({{90}^{o}}-\theta )=\sin \theta$

(c) $\tan ({{90}^{o}}-\theta )=\cot \theta$

(d) $\cot ({{90}^{o}}-\theta )=\tan \theta$

(e) $\sec ({{90}^{o}}-\theta )=\cos ec\theta$

(f) $co\sec ({{90}^{o}}-\theta )=sec\theta$

• Trigonometric identities: An equation involving trigonometric ratios of angle$\theta$is said to be an identity if it is satisfied for all values of$\theta$.

(a) ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$

(b) ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$

(c) $co{{\sec }^{2}}\theta -{{\cot }^{2}}\theta =1$

• Some equations derived from trigonometric identities:

(a) ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$

$\Rightarrow$ ${{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta$

$\Rightarrow$ $\sin \theta =\sqrt{1-{{\cos }^{2}}\theta }$

Similarly,

${{\cos }^{2}}\theta =1-{{\sin }^{2}}\theta$

$\Rightarrow$ $\cos \theta =\sqrt{1-{{\sin }^{2}}\theta }$

(b) ${{\sec }^{2}}q-{{\tan }^{2}}\theta =1$

$\Rightarrow$ ${{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta$

$\Rightarrow$ $\sec \theta =\sqrt{1+{{\tan }^{2}}\theta }$

Similarly,

${{\tan }^{2}}\theta ={{\sec }^{2}}\theta -1$

$\Rightarrow$ $\tan \theta =\sqrt{{{\sec }^{2}}\theta -1}$

• Some more trigonometric identities:

(a) $\cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta$

(b) $\sin 2\theta =2\sin \theta \cos \theta$

(c) $\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta$

(d) $\frac{{{\tan }^{2}}\theta }{1+{{\tan }^{2}}\theta }=si{{n}^{2}}\theta$

(e) $2{{\cos }^{2}}\theta -1=\frac{-1{{\tan }^{2}}\theta }{1+{{\tan }^{2}}\theta }$

(f) $\tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta }$

• Relations expressing one trigonometric ratios in terms of the other
 $\mathbf{sin\theta }$ $\mathbf{cos\theta }$ $\mathbf{tan\theta }$ $\mathbf{cosec\theta }$ $\mathbf{sec\theta }$ $\mathbf{cot\theta }$ $\mathbf{sin\theta }$ $\sin \theta$ $\sqrt{1-{{\cos }^{2}}\theta }$ $\frac{1}{\sqrt{1-{{\tan }^{2}}\theta }}$ $\frac{1}{\cos ec\theta }$ $\frac{\sqrt{{{\sec }^{2}}\theta -1}}{\sec \theta }$ $\frac{1}{\sqrt{1+{{\cot }^{2}}\theta }}$ $\mathbf{cos\theta }$ $\sqrt{1-{{\sin }^{2}}\theta }$ $\cos \theta$ $\frac{1}{\sqrt{1+{{\tan }^{2}}\theta }}$ $\frac{\sqrt{\cos e{{c}^{2}}\theta -1}}{\cos ec\theta }$ $\frac{1}{\sec \theta }$ $\frac{\cot \theta }{\sqrt{1+{{\cot }^{2}}\theta }}$ $\mathbf{tan\theta }$ $\frac{\sin \theta }{\sqrt{1-{{\sin }^{2}}\theta }}$ $\frac{\sqrt{1-{{\cos }^{2}}\theta }}{{{\cos }^{2}}\theta }$ $\tan \theta$ $\frac{1}{\sqrt{\cos e{{c}^{2}}\theta -1}}$ $\sqrt{{{\sec }^{2}}\theta -1}$ $\frac{1}{\cot \theta }$ $\mathbf{cosec\theta }$ $\frac{1}{\sin \theta }$ $\frac{1}{\sqrt{1-{{\cos }^{2}}\theta }}$ $\frac{\sqrt{1+{{\tan }^{2}}\theta }}{\tan \theta }$ $\cos ec\theta$ $\frac{\sec \theta }{\sqrt{{{\sec }^{2}}\theta -1}}$ $\sqrt{1+{{\cot }^{2}}\theta }$ $\mathbf{sec\theta }$ $\frac{1}{\sqrt{1-{{\sin }^{2}}\theta }}$ $\frac{1}{\cos \theta }$ $\sqrt{1+{{\tan }^{2}}\theta }$ $\frac{\sqrt{1+{{\tan }^{2}}\theta }}{\tan \theta }$ $\sec \theta$ $\frac{\sqrt{1+{{\cot }^{2}}\theta }}{\cot \theta }$ $\mathbf{cot\theta }$ $\frac{\sqrt{1-{{\sin }^{2}}\theta }}{\sin \theta }$ $\frac{\cos \theta }{\sqrt{1-{{\cos }^{2}}\theta }}$ $\frac{1}{\tan \theta }$ $\sqrt{\cos e{{c}^{2}}\theta -1}$ $\frac{1}{\sqrt{{{\sec }^{2}}\theta -1}}$ $\cot \theta$

##### Notes - Introduction to Trigonometry

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