Problem 51 - Precalculus Essentials

Question

Use a sketch to find the exact value of each expression. $$\tan \left[\sin ^{-1}\left(-\frac{3}{5}\right)\right]$$

Step-by-Step Solution

Answer

Substitute a value after calculation from step 2 to the equation in step 3 and find the final answer. The value obtained is the exact value of the expression \(\tan(\sin^{-1}(-\frac{3}{5}))\).

Step 1: Identify the sin inverse value

Let's denote the angle \(\sin^{-1}(-\frac{3}{5})\) as \(\theta\) which implies that \(\sin(\theta) = -\frac{3}{5}\). The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse. In this scenario, we can consider -3 as length of the opposite side (as it's negative indicating that it's in the opposite direction), and 5 as the length of the hypotenuse.

Step 2: Calculate the length of the adjacent side

To find the tangent of the angle, we'll need also the length of the adjacent side. This can be obtained by the Pythagorean Theorem which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, \(a = \sqrt{ h^2 - o^2 }\) where h = 5 is the hypotenuse, o = -3 is the opposite side, and a is the adjacent side we're looking for. Calculate the corresponding value.

Step 3: Calculate the tangent of the angle

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case, we already have all values so perform \(\tan(\theta)= \frac{-3}{a}\) calculation to find exact value of the given expression.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, often denoted as 'arc' functions (like arcsin, arccos, and arctan), are used to find the angles when the value of the trigonometric function is known. They're the opposite of regular trigonometric functions. For instance, while the sine function gives you the ratio of the opposite side to the hypotenuse of a right-angled triangle, its inverse function, denoted as \( \sin^{-1} \), gives you the angle when this ratio is known.

In the given exercise, \( \sin ^{-1}\left(-\frac{3}{5}\right) \) is used to represent the angle whose sine is \(-\frac{3}{5}\). Since values of sine function range from -1 to 1, the angle resulting from \( \sin^{-1} \) will always be in the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) for real numbers. Understanding inverse trigonometric functions is essential as they help solve equations and enable the interpretation of a ratio as an angle.

Pythagorean Theorem

The Pythagorean Theorem is a foundational concept in trigonometry and geometry. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be represented as \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the two other sides.

In our problem, once we know the sine of an angle, we can find the hypotenuse and the opposite side, but we need the theorem to solve for the length of the adjacent side. This relationship is crucial in trigonometry, as it bridges the gap between angles and side lengths in right triangles.

Tangent Ratio

The tangent ratio is another trigonometric function like sine and cosine, and it's defined for a particular angle within a right-angled triangle. Tangent specifically refers to the ratio of the length of the opposite side to the length of the adjacent side, and it's represented as \( \tan(\theta) = \frac{o}{a} \). The tangent function is especially useful for measuring steepness or the inclination of an angle.

In this exercise, to find the tangent of the angle represented by \( \sin^{-1}\left(-\frac{3}{5}\right) \), you must first establish the lengths of the opposite and adjacent sides of the associated triangle. With both lengths determined using inverse trigonometric functions and the Pythagorean theorem, you can easily calculate the tangent, which encapsulates the triangle's ratio and ultimately the steepness of the angle we're examining.