2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

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Multiple Choice

If $\tan\theta=\frac{12}{5}$ , find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{5},\csc\theta=\frac{13}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{12},\csc\theta=\frac{13}{5}$

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{5},\csc\theta=-\frac{13}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{12},\csc\theta=-\frac{13}{5}$

Verified step by step guidance

Start by understanding that $ \tan\theta = \frac{12}{5} $ represents the ratio of the opposite side to the adjacent side in a right triangle. This means the opposite side is 12 units and the adjacent side is 5 units.

Use the Pythagorean theorem to find the hypotenuse. The formula is $ c = \sqrt{a^2 + b^2} $, where $ a = 12 $ and $ b = 5 $. Calculate $ c = \sqrt{12^2 + 5^2} $.

Now that you have the hypotenuse, you can find $ \sin\theta $ and $ \cos\theta $. $ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} $ and $ \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} $. Substitute the values to find $ \sin\theta $ and $ \cos\theta $.

To find $ \cot\theta $, use the reciprocal of $ \tan\theta $. Since $ \tan\theta = \frac{12}{5} $, $ \cot\theta = \frac{5}{12} $.

Finally, find $ \sec\theta $ and $ \csc\theta $ using the reciprocals of $ \cos\theta $ and $ \sin\theta $ respectively. $ \sec\theta = \frac{1}{\cos\theta} $ and $ \csc\theta = \frac{1}{\sin\theta} $. Rationalize the denominators if necessary.