Trigonometry 9

Exercise 2.2 (A)

1. Fill in the blanks:

1. $\csc\theta$ in terms of $\sin\theta = \dots\dots$
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   $\csc\theta = \frac{1}{\sin\theta}$
2. $\tan\theta$ in terms of $\cot\theta = \dots\dots$
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   $\tan\theta = \frac{1}{\cot\theta}$
3. $\cos\theta$ in terms of $\sec\theta = \dots\dots$
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   $\cos\theta = \frac{1}{\sec\theta}$
4. $\cot\theta$ in terms of $\tan\theta = \dots\dots$
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   $\cot\theta = \frac{1}{\tan\theta}$
5. $\sin\theta$ in terms of $\csc\theta = \dots\dots$
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   $\sin\theta = \frac{1}{\csc\theta}$
6. $\sec\theta$ in terms of $\cos\theta = \dots\dots$
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   $\sec\theta = \frac{1}{\cos\theta}$
7. $\cot\theta$ in terms of $\sin\theta$ and $\cos\theta = \dots\dots$
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   $\cot\theta = \frac{\cos\theta}{\sin\theta}$
8. $\tan\theta$ in terms of $\sin\theta$ and $\cos\theta = \dots\dots$
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   $\tan\theta = \frac{\sin\theta}{\cos\theta}$

2. Write the quotient relation of trigonometric ratio.

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$$\tan\theta = \frac{\sin \theta}{\cos \theta}$$ $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$

3. Write any three trigonometric identities.

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Any three:

$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$

4. Prepare the list of possible trigonometric relations obtained from Pythagoras theorem.

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$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$ $$\sec^2\theta - \tan^2\theta = 1$$ $$\csc^2\theta - \cot^2\theta = 1$$

5. Simplify:

1. $\tan\theta + 3\tan\theta$
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   $= 4\tan\theta$
2. $4\cot A + 6\cot A + 2\cot A$
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   $= (4+6+2)\cot A$
   $= 12\cot A$