Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\sin \theta=\frac{3}{5}$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{3}{5} \), \( \cos \theta = \frac{4}{5} \), \( \tan \theta = \frac{3}{4} \).

Step 1: Recall the Pythagorean Identity

The Pythagorean identity states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). We will use this identity to find \( \cos \theta \) given \( \sin \theta = \frac{3}{5} \).

Step 2: Substitute the Sin Value

Substitute \( \sin \theta = \frac{3}{5} \) into the Pythagorean Identity: \( \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \).

Step 3: Simplify the Equation

Calculate the square of \( \frac{3}{5} \), which gives \( \frac{9}{25} \). Substitute back into the equation to get \( \frac{9}{25} + \cos^2 \theta = 1 \).

Step 4: Solve for \( \cos^2 \theta \)

Subtract \( \frac{9}{25} \) from 1 to isolate \( \cos^2 \theta \). This results in \( \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \).

Step 5: Find \( \cos \theta \)

Take the square root of both sides to find \( \cos \theta \). Since \( \theta \) is an acute angle, \( \cos \theta \) is positive. Thus, \( \cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \).

Step 6: Find \( \tan \theta \)

To find \( \tan \theta \), use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute \( \sin \theta = \frac{3}{5} \) and \( \cos \theta = \frac{4}{5} \) into this formula to get \( \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), the sum of the squares of the sine and cosine of that angle equals one. Mathematically, this can be expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is named after the Pythagorean theorem, due to the similarity between the relationships. It is an essential tool in trigonometry, providing a connection between sine and cosine. It is very useful when you know either \( \sin \theta \) or \( \cos \theta \) and need to find the other. For this exercise, knowing \( \sin \theta = \frac{3}{5} \) allows us to apply the Pythagorean identity to solve for \( \cos \theta \).

sin theta

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. In mathematical terms, if you have an angle \( \theta \), then:

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

In our exercise, \( \sin \theta = \frac{3}{5} \). This means that if you consider a right triangle where the hypotenuse is 5 units long, the length of the side opposite the angle \( \theta \) is 3 units. Understanding sine is crucial for solving various problems in trigonometry, as it relates the angle with the sides of the triangle.

cos theta

The cosine function relates an angle \( \theta \) to the adjacent side and hypotenuse of a right triangle. It is defined as:

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

This is particularly helpful when calculating angles or side lengths in triangles. In the exercise, the value of \( \cos \theta \) is derived using the Pythagorean identity, given \( \sin \theta = \frac{3}{5} \). Next, we solve for \( \cos \theta \) by rearranging the identity to \( \cos^2 \theta = 1 - \sin^2 \theta \). This results in \( \cos^2 \theta = \frac{16}{25} \), and taking the positive square root due to the acute angle, we find \( \cos \theta = \frac{4}{5} \).

tan theta

The tangent of an angle \( \theta \) is another essential trigonometric function, representing the ratio of the opposite side to the adjacent side of a right triangle. It can also be expressed in terms of sine and cosine as:

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

In the context of this exercise, we are given \( \sin \theta = \frac{3}{5} \) and have calculated \( \cos \theta = \frac{4}{5} \). With these values, we find \( \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \). Understanding tangent is vital for solving problems that involve right triangles and circular functions, as it connects the sine and cosine functions into one comprehensive ratio.