Problem 19 - Precalculus: Functions and Graphs

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: Understanding the Tangent Ratio

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, \(\tan \theta = \frac{5}{12}\), which means the length of the opposite side is 5 and the length of the adjacent side is 12.

Step 2: Use the Pythagorean Theorem to Find the Hypotenuse

To find the exact values of all trigonometric functions, we need the hypotenuse. Using the Pythagorean theorem, we have \(c^2 = a^2 + b^2\), where \(a = 5\) (opposite side) and \(b = 12\) (adjacent side). We get \(c^2 = 5^2 + 12^2 = 25 + 144 = 169\). Therefore, \(c = \sqrt{169} = 13\).

Step 3: Calculate Sine and Cosine

The sine of an angle is the ratio of the opposite side to the hypotenuse: \(\sin \theta = \frac{5}{13}\). The cosine of an angle is the ratio of the adjacent side to the hypotenuse: \(\cos \theta = \frac{12}{13}\).

Step 4: Summary of Trigonometric Functions

We have found the exact values for the principal trigonometric functions of the angle \(\theta\). They are: \(\tan \theta = \frac{5}{12}\), \(\sin \theta = \frac{5}{13}\), and \(\cos \theta = \frac{12}{13}\).

Key Concepts

Tangent Ratio

The tangent ratio in trigonometry relates to how you find the tangent of an angle in a right triangle. Tangent, often abbreviated as tan, is the ratio of the length of the opposite side to the length of the adjacent side. In simpler terms, for any angle \( \theta \) in a right triangle:

• Opposite side is the side opposite to the angle \( \theta \).
• Adjacent side is the side next to the angle \( \theta \), but not the hypotenuse.

If you have the tangent of an angle, say \( \tan \theta = \frac{5}{12} \), you know that for every 5 units of height (opposite), there are 12 units in length (adjacent).
This ratio provides a simple way to connect angles with measurements, making it easier to solve triangles once you know any one part of a right triangle.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental relationship in geometry among the three sides of a right triangle. It is expressed as \( c^2 = a^2 + b^2 \), where:

• \( c \), the hypotenuse, is opposite the right angle and is the longest side of the triangle.
• \( a \) and \( b \) are the two other sides, known as the legs of the triangle.

In our exercise, given \( a = 5 \) (opposite side) and \( b = 12 \) (adjacent side), applying the Pythagorean theorem gives:
\[ c^2 = 5^2 + 12^2 = 25 + 144 = 169 \]Thus, the hypotenuse \( c = \sqrt{169} = 13 \).
Knowing the length of all three sides in a right triangle is invaluable for calculating other trigonometric functions, like sine and cosine, of the related angle.

Sine and Cosine

Sine and cosine are two key functions in trigonometry, like tangent, which help describe relationships in right triangles. They are both based on the hypotenuse of the triangle, which is the longest side.
The sine of an angle \( \theta \) is the ratio of the opposite side to the hypotenuse:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

In our example, \( \sin \theta = \frac{5}{13} \), meaning the side opposite \( \theta \) is 5, and the hypotenuse is 13.
The cosine of an angle \( \theta \) is the ratio of the adjacent side to the hypotenuse:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

For \( \cos \theta = \frac{12}{13} \), it shows that the adjacent side is 12, while the hypotenuse remains 13.
Sine and cosine are crucial for solving various problems in trigonometry and are widely used in fields like physics and engineering.