Trigonometric ratios of angles associated to positive acute angle(theta)

Trigonometric Ratios of Angles Associated with a Positive Acute Angle θ

In trigonometry, angles greater than 90° or negative angles are often simplified using reference angles, also known as associated angles. These reference angles are acute (less than 90°) and help in evaluating the trigonometric ratios for any given angle using the known values of θ in the first quadrant.

What Are Associated Angles?

Associated or reference angles are angles made between the terminal side of a given angle and the x-axis. They help simplify trigonometric calculations by converting any angle to its first-quadrant equivalent while adjusting the sign based on the quadrant.

Steps to Find Trigonometric Ratios of Associated Angles

1. Step 1: Identify the given angle and its quadrant.
2. Step 2: Find the reference (acute) angle θ it corresponds to by subtracting from 180°, 360°, etc., as required.
3. Step 3: Refer to the trigonometric ratio corresponding to the acute angle θ.
4. Step 4: Determine the sign of the function based on the quadrant where the original angle lies.

Mnemonic to Remember Signs in Quadrants

Use the phrase “All Students Take Calculus” to remember:

• 1st Quadrant (All): All trigonometric functions are positive.
• 2nd Quadrant (Students): Sine and cosecant are positive.
• 3rd Quadrant (Take): Tangent and cotangent are positive.
• 4th Quadrant (Calculus): Cosine and secant are positive.

Example

Let’s find sin(150°):

• 150° lies in the 2nd quadrant.
• Reference angle = 180° − 150° = 30°.
• In the second quadrant, sin(150°) equals sin(30°) = 1/2 because sine remains positive.

General Angle Relationships

Using associated angles, the trigonometric identities can be written as:

• sin(180° − θ) = sin θ
• cos(180° − θ) = −cos θ
• tan(180° − θ) = −tan θ
• sin(360° − θ) = −sin θ
• cos(360° − θ) = cos θ
• tan(360° − θ) = −tan θ

This method makes it easier to compute trigonometric values for angles in any quadrant by reducing them to acute angles and then applying appropriate signs.