# 9th Class Mathematics Trigonometrical Ratios and Identities Trigonometrical Identities

## Trigonometrical Identities

Category : 9th Class

### Trigonometrical Identities

In the adjoining figure triangle DEF is a right angled triangle right angle at D. Then the trigonometrical identities are

1. ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$

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

3. $co{{\sec }^{2}}\theta -{{\cot }^{2}}\theta =1$

We can also derive different relations between identities in different form

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

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

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

Similarly

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

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

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

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

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

Similarly

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

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

(c) $\cos e{{c}^{2}}\theta -{{\cot }^{2}}\theta =1$

or            $\cos e{{c}^{2}}\theta =1+{{\cot }^{2}}\theta$

or            $\cos e{{c}^{2}}\theta =\pm \sqrt{1+{{\cot }^{2}}\theta }$

Similarly

${{\cot }^{2}}\theta =\cos e{{c}^{2}}\theta -1$

or $\cot \theta =\pm \sqrt{\cos e{{c}^{2}}\theta -1}$

Verification of $\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{\theta +co}{{\mathbf{s}}^{\mathbf{2}}}\mathbf{\theta =1}$

Suppose $\Delta \text{RST}$ is a right angled triangle in which right angle at R and angle S in formed at$\theta$.

Here, perpendicular RT represented by "p", base RS represented by "b" and hypotenuse "ST" represented by h.

Then by Pythagoras theorem,

${{p}^{2}}+{{b}^{2}}={{h}^{2}}$

As we know

$\sin \theta =\frac{p}{h},$ $\cos \theta =\frac{b}{h}$

Therefore,

${{\sin }^{2}}\theta =\frac{{{p}^{2}}}{{{h}^{2}}}\,\,\text{and}\,\,{{\cos }^{2}}\theta =\frac{{{b}^{2}}}{{{h}^{2}}}$

${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =\frac{{{p}^{2}}}{{{h}^{2}}}+\frac{{{b}^{2}}}{{{h}^{2}}}=\frac{{{p}^{2}}+{{b}^{2}}}{{{h}^{2}}}=\frac{{{h}^{2}}}{{{h}^{2}}}=1$

Therefore, ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$

Similarly we can verify other results.

Values of Different Trigonometrical Ratios in Different Quadrants

Above shows the different quadrants. The following are about the T- Ratios when angles lies in different quadrants

1.    When the angle lies in the I quadrant all trigonometrical ratios are positive i.e. The value of $\text{sin}\theta ,\text{ cos}\theta ,\text{ tan}\theta ,\text{ cot}\theta ,\text{ sec}\theta$and $\text{cosec}\theta$ are positive.

2.    When the angle$''\theta ''$ lies in the second quadrant, the value of sine and cosec are positive and other ratios like $\text{cos}\theta ,\text{ tan}\theta ,\text{ cot}\theta$ and $\text{sec}\theta$ are negative.

3.    When an angle $''\theta ''$ lies in third quadrant, the value of tan e and cote are positive and other ratios like $\text{sin}\theta \text{,}\,\text{cos}\theta ,\text{ cosec}\theta$and $\sec \theta$ are negative.

4.    When an angle $''\theta ''$ lies in fourth quadrant, the value of $\text{cos}\theta \text{ sec}\theta$ are positive and other ratios like $\text{sin}\theta ,\text{ tan}\theta ,\text{ cosec}\theta$ and cote are negative.

From the figure given below you can remember it easily

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