Question

Find the reference angle of each angle. $$ -165^{\circ} $$

Short answer

The reference angle is 15°.

Step-by-step solution

Add 360° to the Negative Angle

To convert a negative angle to a positive angle within one full rotation (0° to 360°), add 360° to the given angle. Calculate the positive equivalent of the angle by adding 360° to -165°.\[-165° + 360° = 195°\]

Identify the Quadrant

Determine which quadrant the positive angle lies in by analyzing the resulting angle. Since 195° is between 180° and 270°, it lies in the third quadrant.

Find the Reference Angle

In the third quadrant, the reference angle is found by subtracting the angle from 180°. Calculate the reference angle as follows:\[195° - 180° = 15°\]

Key concepts

Negative Angles

When dealing with angles in trigonometry, you may come across negative angles. Negative angles are measured in the clockwise direction from the positive x-axis. If you have a negative angle, like \( -165^{\circ} \), you can convert it to a positive angle between \( 0^{\circ} \) and \( 360^{\circ} \) by adding \( 360^{\circ} \). This adjustment will provide an equivalent positive angle that is easier to work with. In our example:\[-165^{\circ} + 360^{\circ} = 195^{\circ}\]Understanding this concept helps you analyze and simplify trigonometry problems.

Quadrants

The coordinate plane is divided into four quadrants, each representing different ranges of angle measures:

• Quadrant I: \(0^{\circ} \) to \(90^{\circ} \)
• Quadrant II: \(90^{\circ} \) to \(180^{\circ} \)
• Quadrant III: \(180^{\circ} \) to \(270^{\circ} \)
• Quadrant IV: \(270^{\circ} \) to \(360^{\circ} \)

For our positive angle \( 195^{\circ} \), it falls within the range \( 180^{\circ} \) to \( 270^{\circ} \), placing it in the third quadrant. This information helps determine how the angle interacts with the axes and which trigonometric functions will be positive or negative.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between angles and sides of triangles, often using circles to understand these relationships better. One crucial concept is the reference angle, which is the smallest angle between the terminal side of the given angle and the x-axis.To find the reference angle in the third quadrant, subtract the angle from \( 180^{\circ} \). For example, with \( 195^{\circ} \), the calculation is:\[195^{\circ} - 180^{\circ} = 15^{\circ} \]This gives the reference angle, which can then be used to find trigonometric values efficiently. Understanding reference angles helps simplify problems and makes it easier to work with sine, cosine, and tangent functions.