Math Problem Statement

Find an angle θ with 0° < θ < 360° that has the same sine and cosine function values as 240°.

Solution

The question asks to find an angle θ\theta such that 0<θ<3600^\circ < \theta < 360^\circ, which has:

  1. The same sine value as 240240^\circ.
  2. The same cosine value as 240240^\circ.

1. Finding θ\theta with the same sine value as 240240^\circ

The sine function is periodic, and for a given angle, another angle can share the same sine value within one full revolution. The reference angle for 240240^\circ is 6060^\circ (since 240180=60240^\circ - 180^\circ = 60^\circ).

Since sine is negative in the third and fourth quadrants, we need to find the angle in the fourth quadrant that has the same sine value as 240240^\circ.

The sine of 240240^\circ is negative, and another angle that has the same sine value is 360240=120360^\circ - 240^\circ = 120^\circ.

Thus, the angle with the same sine value as 240240^\circ is 120120^\circ.

2. Cosine value of 240240^\circ

The cosine of 240240^\circ is negative as well, and the angle that has the same cosine value is 120120^\circ, which is already given.


Would you like further explanation on trigonometric identities, or do you have any specific questions?

Related Questions:

  1. How do we find reference angles for trigonometric functions?
  2. What is the general solution for angles with the same sine value in trigonometry?
  3. Can an angle have the same sine and cosine values?
  4. Why is the sine of angles in the third and fourth quadrants negative?
  5. How do we use the unit circle to find angles with the same sine or cosine values?

Tip:

Always use the unit circle to check the signs of sine and cosine for angles in different quadrants—it helps in finding equivalent angles!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Function
Cosine Function
Reference Angles
Unit Circle

Formulas

sin(θ) = sin(180° + α)
cos(θ) = cos(360° - α)

Theorems

Properties of Trigonometric Functions
Periodic Nature of Sine and Cosine

Suitable Grade Level

Grades 10-12