Problem 51 - Algebra and Trigonometry

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

The exact value of the expression \(\tan [\sin ^{-1}(-\frac{3}{5})]\) is -0.75

Step 1: Determine the Angle from Inverse Sine Function

Since \(\sin ^{-1} (-\frac{3}{5})\) represents an angle whose sine is \(-\frac{3}{5}\), the opposite side is -3 (considering sine as a negative value indicates that the angle is located in the 3rd or 4th quadrant) and the hypotenuse is 5. This refers to a right triangle.

Step 2: Calculate the Adjacent Side Using Pythagoras’ Theorem

We can calculate the length of the adjacent side of the triangle using the Pythagorean theorem: \(a^2 + b^2 = c^2\), where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. Let a be the adjacent side, then we can solve \(a = \sqrt{c^2 - b^2} = \sqrt{5^2 - (-3)^2} = 4\).

Step 3: Find the Tangent of the Angle

Now, we can find the tangent of the angle using the definition of tangent as the ratio of the opposite side to the adjacent side: \(\tan(\sin^{-1}(-\frac{3}{5})) = \frac{-3}{4} = -0.75\) in this case.

Key Concepts

Trigonometric Identities

Trigonometric identities are vital tools in trigonometry. They help us connect different trigonometric functions. One key identity relates sine, cosine, and tangent:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

This tells us that tangent is the ratio of sine to cosine.
When dealing with inverse trigonometric functions, these identities simplify calculations. For example, if you know \( \sin^{-1}(x) \), you can use identities to find other functions like \( \tan(\theta) \) or \( \cos(\theta) \).

• Functions like tangent become particularly useful in right triangles.
• Understanding these relationships aids in solving various trigonometric problems.

Right Triangle Properties

Right triangles have distinctive properties making them essential in trigonometry. They consist of:

• One right angle (90 degrees).
• Two other angles that add up to 90 degrees.
• Three sides: opposite, adjacent, and hypotenuse.

In the exercise, we have an angle determined by \( \sin^{-1}(-\frac{3}{5}) \).
This means the triangle's opposite side is -3, indicating direction rather than length, and the hypotenuse is 5.
Using these properties, finding missing side lengths and angle measures becomes easier.

• Right triangles allow application of trigonometric functions to determine unknowns.
• They are foundational in understanding and solving real-world problems.

Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in trigonometry. It connects the sides of a right triangle:

• \( a^2 + b^2 = c^2 \)

Here, \(a\) and \(b\) are the triangle's legs, and \(c\) is the hypotenuse.
In the given exercise, we use this theorem to find the length of the adjacent side.
Knowing any two side lengths allows you to calculate the third.

• This theorem helps in establishing relationships between angles and sides.
• It's crucial for creating models and solving geometric problems.

For example, we found the adjacent side to be 4, given that the hypotenuse was 5 and the opposite was -3. This illustrates the theorem's practical application in trigonometrical setups.