Question

In Exercises 41 and \(42,\) cvaluate the expression. $$\tan \left(\sin ^{-1}\left(\frac{9}{13}\right)\right)$$

Short answer

The result of evaluating the expression \(\tan \left(\sin ^{-1}\left(\frac{9}{13}\right)\right)\) is \(\frac{9}{\sqrt{40}}\).

Step-by-step solution

Understanding the Problem

In this step we need to understand the nature of the expression \(\tan \left(\sin ^{-1}\left(\frac{9}{13}\right)\right)\). Here, \(\sin ^{-1}\left(\frac{9}{13}\right)\) represents an angle whose sine is \(\frac{9}{13}\). This forms the component of an imaginary right triangle where the opposite side is 9 and the hypotenuse is 13.

Determine the Adjacent Side

By Pythagorean theorem the length of the adjacent side is determined by calculating the square root of the difference between the square of the hypotenuse and the square of the opposite side, which means \(\sqrt{13^{2} - 9^{2}} = \sqrt{40}\).

Evaluate the Expression

The original expression \(\tan \left(\sin ^{-1} \left(\frac{9}{13}\right)\right)\) can now be thought as asking the tangent of the angle whose sine is \(\frac{9}{13}\). The tangent of an angle in a right triangle is given by the ratio of the length of the opposite side to the length of the adjacent side. Hence, \(\tan \left(\sin ^{-1}\left(\frac{9}{13}\right)\right) = \frac{9}{\sqrt{40}}\).

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially when dealing with right-angled triangles. It states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

This relationship can be expressed with the equation: \( a^2 + b^2 = c^2 \), where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the triangle's other two sides. By rearranging this formula, we can solve for any side as long as we have the measurements of the other two.

In the context of the textbook exercise, we apply the Pythagorean theorem to find the length of the side adjacent to the angle whose sine is given. If the opposite side (labeled as 'opposite' in reference to the angle) is 9 units and the hypotenuse is 13 units long, the formula becomes:\[ 9^2 + b^2 = 13^2 \]\[ b = \sqrt{13^2 - 9^2} \]\[ b = \sqrt{40} \]
Once the length of the adjacent side is found, it aids in calculating trigonometric functions related to the angle in question.

Tangent Function

The tangent function, usually abbreviated as 'tan', is one of the six fundamental trigonometric functions. In the context of right triangle trigonometry, it relates an angle of the triangle to the ratio of the triangle's opposite side to its adjacent side. The formula for tangent is given by:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \].

What is special about the tangent function compared to other trigonometric functions is that it deals with the ratio of two sides, making it very useful in instances where you need to find the slope of a line or the angle of elevation and depression.

For the given expression \( \tan(\sin^{-1}(\frac{9}{13}) \) from our exercise, once we find the adjacent side using the Pythagorean theorem, the tangent of the angle can be calculated directly. With the opposite side being 9 and the adjacent side being the square root of 40, the tangent of the angle is just the ratio of these two lengths:\[ \tan(\theta) = \frac{9}{\sqrt{40}} \].

Right Triangle Trigonometry

Right triangle trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of right triangles. We commonly use trigonometric functions like sine, cosine, and tangent to describe these relationships.

The sine of an angle, denoted by 'sin', is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine, 'cos', is the ratio of the length of the adjacent side to the hypotenuse. As explained previously, the tangent, 'tan', is the ratio of the length of the opposite side to the length of the adjacent side.

When given an angle measure, such as the inverse sine (\( \sin^{-1} \)) of a value, we're being asked to find the angle whose sine is that value. In the case of \( \sin^{-1}(\frac{9}{13}) \), we're looking for the angle at which the opposite side is 9 units, and the hypotenuse is 13 units long in a right triangle setup. Once that angle is found, determining the other trigonometric functions of that angle, like its tangent as solved in our exercise, becomes straightforward.