91Ó°ÊÓ

When solving trigonometric equations, the objective is to find the values of the variable that make the equation true. In our example, we are given \(\frac{1}{3}\theta = 45^{\bullet}, 60^{\bullet}, 75^{\bullet}, 90^{\bullet}\). Here are the steps:

• First, we need to isolate \(\theta\) by multiplying both sides of the equation by 3.
• Body: The overall goal is to simplify each expression to find the specific values of \(\theta\).

This process can be broken into simple steps:
\(\frac{1}{3}\theta = 45^{\bullet}\) becomes \(\theta = 3 \times 45^{\bullet} = 135^{\bullet}\)
\(\frac{1}{3}\theta = 60^{\bullet}\) becomes \(\theta = 3 \times 60^{\bullet} = 180^{\bullet}\)
\(\frac{1}{3}\theta = 75^{\bullet}\) becomes \(\theta = 3 \times 75^{\bullet} = 225^{\bullet}\)
\(\frac{1}{3}\theta = 90^{\bullet}\) becomes \(\theta = 3 \times 90^{\bullet} = 270^{\bullet}\)

By following these straightforward steps, you can easily solve similar trigonometric equations with multiplication.

Theta (\(\theta\)) represents an angle, often measured in degrees or radians. In the context of trigonometric equations, you will find \(\theta\) as part of the equations needing solutions. In the given exercise, we are focusing on degrees.

Each \(\theta\) value must be computed within the specified interval \([0^{\bullet}, 360^{\bullet})\). In our case, \(\frac{1}{3}\theta = 45^{\bullet}, 60^{\bullet}, 75^{\bullet}, 90^{\bullet}\), \(\theta\) is isolated by multiplying by 3. The solutions \(\theta = 135^{\bullet}, 180^{\bullet}, 225^{\bullet}, 270^{\bullet}\) fall within the given interval and are correct.

Key points to remember:

• Make sure your solutions fall within the given interval.
• Always double-check your final \(\theta\) values to ensure accuracy.

Understanding intervals and degree measures are crucial for mastering trigonometric equations.

Multiplication can simplify trigonometric equations by isolating the variable. Here’s why it works:

Given the equation \(\frac{1}{3}\theta = 45^{\bullet}\), we multiply both sides by 3, which reverses the division by 3:

\[ 3 \times \frac{1}{3}\theta = 3 \times 45^{\bullet} \]
Simplifying, we get \(\theta = 135^{\bullet}\)

The same approach applies to the other parts of the exercise:

\(\frac{1}{3}\theta = 60^{\bullet}\) results in \(\theta = 180^{\bullet}\)
\(\frac{1}{3}\theta = 75^{\bullet}\) results in \(\theta = 225^{\bullet}\)
\(\frac{1}{3}\theta = 90^{\bullet}\) results in \(\theta = 270^{\bullet}\)

Multiplying both sides by the same number (in this case 3) simplifies the equations and helps in isolating \(\theta\). Understanding this concept is essential for solving more complex trigonometric equations effectively.