Ratios & Identities: Trigonometric Ratios of Allied Angles

Trigonometric ratios of allied angles are an important concept in trigonometry. Allied angles are angles that differ by an integer multiple of 90 degrees (or π/2 radians). Understanding the trigonometric ratios of these angles is crucial for solving various problems in trigonometry, including those involving wave functions, oscillations, and rotations.

Basic Trigonometric Ratios

Before we delve into allied angles, let's review the basic trigonometric ratios for an angle θ in a right-angled triangle:

• Sine (sin): $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$
• Cosine (cos): $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$
• Tangent (tan): $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$
• Cosecant (csc): $\csc(\theta) = \frac{1}{\sin(\theta)}$
• Secant (sec): $\sec(\theta) = \frac{1}{\cos(\theta)}$
• Cotangent (cot): $\cot(\theta) = \frac{1}{\tan(\theta)}$

Trigonometric Ratios of Allied Angles

When dealing with allied angles, we use the sum or difference of an angle with a multiple of 90 degrees. The trigonometric ratios for these angles can be determined using certain identities and the concept of reference angles.

Here's a table summarizing the trigonometric ratios for allied angles:

Angle	Sine	Cosine	Tangent	Cosecant	Secant	Cotangent
θ	sin(θ)	cos(θ)	tan(θ)	csc(θ)	sec(θ)	cot(θ)
90° ± θ	cos(θ)	∓ sin(θ)	∓ cot(θ)	sec(θ)	∓ csc(θ)	∓ tan(θ)
180° ± θ	∓ sin(θ)	-cos(θ)	∓ tan(θ)	∓ csc(θ)	-sec(θ)	∓ cot(θ)
270° ± θ	∓ cos(θ)	± sin(θ)	± cot(θ)	∓ sec(θ)	± csc(θ)	± tan(θ)
360° ± θ	sin(θ)	cos(θ)	tan(θ)	csc(θ)	sec(θ)	cot(θ)

The signs in the table depend on the quadrant in which the allied angle lies. The sign is positive if the function is positive in that quadrant and negative otherwise.

Formulas for Allied Angles

The trigonometric ratios for angles that are sums or differences of an angle θ and a multiple of 90° can be derived using the following formulas:

1. $\sin(90° ± θ) = \cos(θ)$
2. $\cos(90° ± θ) = \mp \sin(θ)$
3. $\tan(90° ± θ) = \mp \cot(θ)$
4. $\csc(90° ± θ) = \sec(θ)$
5. $\sec(90° ± θ) = \mp \csc(θ)$
6. $\cot(90° ± θ) = \mp \tan(θ)$

Examples

Let's look at some examples to understand how to apply these formulas:

Example 1: Find the sine of 150°.

150° is 180° - 30°, so we use the formula for sine:

$\sin(150°) = \sin(180° - 30°) = -\sin(30°)$

Since sine is negative in the second quadrant, we get:

$\sin(150°) = -\frac{1}{2}$

Example 2: Find the tangent of 225°.

225° is 180° + 45°, so we use the formula for tangent:

$\tan(225°) = \tan(180° + 45°) = \tan(45°)$

Since tangent is positive in the third quadrant, we get:

$\tan(225°) = 1$

Example 3: Find the secant of 300°.

300° is 360° - 60°, so we use the formula for secant:

$\sec(300°) = \sec(360° - 60°) = \sec(60°)$

Since secant is positive in the fourth quadrant, we get:

$\sec(300°) = 2$

Understanding the trigonometric ratios of allied angles is essential for simplifying expressions and solving equations in trigonometry. By using the reference angle and the quadrant information, we can easily find the values of trigonometric functions for angles that are not standard angles.