Problem 28 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \tan \left(\sin ^{-1} \frac{4}{5}\right) $$

Step-by-Step Solution

Answer

The exact value of \( \tan \left( \sin^{-1} \frac{4}{5} \right) \) is \( \frac{4}{3} \).

Step 1: Understand the Problem

We need to find the exact value of \( \tan(\sin^{-1}(\frac{4}{5})) \). This involves understanding the relationship between sine, arc sine, and tangent functions.

Step 2: Use Inverse Sine Definition

Let \( \theta = \sin^{-1}\left(\frac{4}{5}\right) \). It means that \( \sin \theta = \frac{4}{5} \), and we need to find \( \tan \theta \).

Step 3: Construct a Right Triangle

Construct a right triangle with angle \( \theta \), where the opposite side is 4 (numerator of the sine fraction), and the hypotenuse is 5 (denominator of the sine fraction).

Step 4: Find the Adjacent Side

Use the Pythagorean theorem \( a^2 + b^2 = c^2 \) to find the adjacent side (\( a \)). Substitute \( a = 4 \) and \( c = 5 \) to get:\[ a^2 + 4^2 = 5^2 \] \[ a^2 + 16 = 25 \] \[ a^2 = 9 \] \[ a = 3 \].

Step 5: Calculate the Tangent Value

Tangent of angle \( \theta \) is the ratio of the opposite to the adjacent side in the triangle. Thus, \( \tan \theta = \frac{4}{3} \).

Step 6: Verify Conclusion

Since all steps follow the logic of trigonometry functions, the calculated \( \tan \theta \) value is correct.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given a trigonometric ratio. Unlike regular trigonometric functions that give us the ratio from an angle, inverse functions work inversely. They help us decode these ratios back into an angle. These functions include arcsine (\( \sin^{-1} \)), arccosine (\( \cos^{-1} \)), and arctangent (\( \tan^{-1} \)).

When you see something like \( \sin^{-1} \left(\frac{4}{5}\right) \), you are being asked to find the angle whose sine is \( \frac{4}{5} \).

• The arcsine output is an angle, usually in radians or degrees.
• Arcsine’s range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

By finding \( \theta = \sin^{-1}\left(\frac{4}{5}\right) \), we establish an angle \( \theta \) whose sine is \( \frac{4}{5} \), readying us to compute further trigonometric values with it.

Right Triangle Trigonometry

Right triangle trigonometry involves analyzing triangles where one angle is 90 degrees. This understanding is fundamental to solving many trigonometric problems.

For a given angle in a right triangle:

• The opposite side is the side opposite the angle.
• The adjacent side is next to the angle and is part of forming it.
• The hypotenuse is the longest side, opposite the right angle.

The essence of this concept is wrapped around these core definitions.

In our example problem, constructing a right triangle with angle \( \theta \) helps visualize its trigonometric ratios. If \( \sin \theta = \frac{4}{5} \), then the triangle has an opposite side of length 4 and a hypotenuse of length 5. Using these attributes, we can swiftly derive other trigonometric values for that angle.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry used to relate the sides of a right triangle. It states:\[ a^2 + b^2 = c^2 \]where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse.

In our example problem, knowing two sides of the right triangle allows us to calculate the third. For instance, given the opposite side as 4 and the hypotenuse as 5, we can find the adjacent side using:

• \( a^2 + 4^2 = 5^2 \)
• Simplifying gives: \( a^2 + 16 = 25 \)
• Then, \( a^2 = 9 \) and so \( a = 3 \)

This calculation provides all necessary sides to determine the tangent value of the angle, reinforcing the relationship between all three sides. The Pythagorean Theorem is not just a formula; it is a key tool in solving complex trigonometric expressions by grounding every calculation in geometric reality.