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Exercises Trigonometry

Example 8 from Algebra and Trigonometry, 8.3 Inverse Trigonometric Functions

Evaluating the Composition of a Sine with an Inverse Tangent

Find an exact value for $\sin (\tan^{- 1} (\frac{7}{4})) .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
While we could use a similar technique as in Example 6, we will demonstrate a different technique here. From the inside, we know there is an angle such that $\tan θ = \frac{7}{4} .$ We can envision this as the opposite and adjacent sides on a right triangle, as shown in Figure 12.

Figure 12 A right triangle with two sides known

Using the Pythagorean Theorem, we can find the hypotenuse of this triangle.

$\begin{array}{l} \begin{array}{l} 4^{2} + 7^{2} = {hypotenuse}^{2} \end{array} \\ hypotenuse = \sqrt{65} \end{array}$
Now, we can evaluate the sine of the angle as the opposite side divided by the hypotenuse.

$\sin θ = \frac{7}{\sqrt{65}}$
This gives us our desired composition.

$\begin{array}{l} \sin (\tan^{- 1} (\frac{7}{4})) = \sin θ \\ = \frac{7}{\sqrt{65}} \\ = \frac{7 \sqrt{65}}{65} \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Evaluate $\cos (\sin^{- 1} (\frac{7}{9})) .$

How did it go?

PreviousEvaluating the Composition of a Sine with an Inverse Cosine Next Finding the Cosine of the Inverse Sine of an Algebraic Expression

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 8.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.