Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\tan \theta=\frac{5}{12}$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{5}{13}, \cos \theta = \frac{12}{13}, \tan \theta = \frac{5}{12} \).

Step 1: Use Trigonometric Identity

We know that the tangent of an angle is the ratio of the opposite side to the adjacent side in a right-angled triangle. Therefore, given \( \tan \theta = \frac{5}{12} \), the opposite side is 5 and the adjacent side is 12.

Step 2: Calculate the Hypotenuse Using Pythagorean Theorem

Use the Pythagorean theorem: \( c^2 = a^2 + b^2 \), where \( a = 5 \) and \( b = 12 \). Solve for \( c \):\[ c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] Thus, the hypotenuse is 13.

Step 3: Find Sine of \( \theta \)

Sine is the ratio of the opposite side to the hypotenuse. Therefore, \( \sin \theta = \frac{5}{13} \).

Step 4: Find Cosine of \( \theta \)

Cosine is the ratio of the adjacent side to the hypotenuse. Therefore, \( \cos \theta = \frac{12}{13} \).

Step 5: Recap the Values

We found that the trigonometric functions for the angle \( \theta \) are:\[ \sin \theta = \frac{5}{13}, \cos \theta = \frac{12}{13}, \tan \theta = \frac{5}{12} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variable. They are incredibly useful as they allow us to express one trigonometric function in terms of another. This can simplify complex problems and help make calculations more manageable.
For this exercise, we make use of the identity for tangent:

• The tangent function, \( \tan \theta \), is defined as the ratio of the opposite side to the adjacent side in a right-angle triangle. This means that \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

From the given \( \tan \theta = \frac{5}{12} \), we identify the opposite side as 5 and the adjacent side as 12 relative to angle \( \theta \).
Knowing these sides allows us to calculate the hypotenuse, and ultimately find the other trigonometric functions such as sine and cosine using these identities.

Right-Angle Triangle

A right-angle triangle is a triangle with one angle measuring 90 degrees. Understanding these triangles is fundamental when working with trigonometric functions because they are the foundation for defining these functions.
For any angle in a right-angle triangle:

• Sine (\( \sin \theta \)) is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)) is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)), as we previously mentioned, is the ratio of the opposite to the adjacent side.

Using the given angle \( \theta \) where \( \tan \theta = \frac{5}{12} \), we visualize a right-angle triangle with the opposite side as 5 and the adjacent side as 12. This setup helps us find the hypotenuse by using the Pythagorean theorem, allowing us to calculate sine and cosine.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of sides in a right-angle triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

• Mathematically, it is represented as \( c^2 = a^2 + b^2 \).

In this problem, we use the Pythagorean theorem to find the hypotenuse of our right-angle triangle. We know:

• Opposite side (\( a \)) = 5
• Adjacent side (\( b \)) = 12

Calculating the hypotenuse:\( c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \) means the hypotenuse is 13. With the hypotenuse and other sides known, we can now easily determine \( \sin \theta = \frac{5}{13} \) and \( \cos \theta = \frac{12}{13} \).
This showcases how the Pythagorean theorem acts as a bridge linking known sides to unknowns in trigonometry problems.