Trigonometry: Compound Angles, Proving Identities & Special Angles

Trigonometric Concepts

1. Compound Angles

Compound angles involve the addition or subtraction of two angles in trigonometric functions. For example, sin(A+B) or cos(A−B). The formulas for these are:

Formulas

• Sine of a sum: sin(A + B) = sinA cosB + cosA sinB
• Sine of a difference: sin(A − B) = sinA cosB − cosA sinB
• Cosine of a sum: cos(A + B) = cosA cosB − sinA sinB
• Cosine of a difference: cos(A − B) = cosA cosB + sinA sinB

Why Do These Formulas Exist?

These formulas come from how sine and cosine are defined on the unit circle, which relates angles to coordinates.

Example Problem

Find sin(75°) using sin(A + B).

1. Break 75° into two known angles, e.g., 45° + 30°.
2. Use the formula: sin(A + B) = sinA cosB + cosA sinB.
3. Substitute: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°).
4. Use known values:
   • sin(45°) = cos(45°) = √2/2
   • sin(30°) = 1/2
   • cos(30°) = √3/2
5. Simplify: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2).
6. Final answer: sin(75°) = √6/4 + √2/4.

2. Proving Identities

Proving trigonometric identities means showing that one side of an equation equals the other, using trigonometric rules.

How to Approach Proving Identities

• Start with one side: Usually, work with the more complicated side.
• Simplify using formulas: Apply compound angle formulas, Pythagorean identities, or reciprocal identities.
• Match the other side: Keep simplifying until both sides look identical.

Example Problem

Prove: sin²A + cos²A = 1.

1. This identity comes from the Pythagorean Theorem on the unit circle.
2. On a unit circle, the equation of the circle is x² + y² = 1.
3. Here, x = cosA and y = sinA.
4. Substitute: cos²A + sin²A = 1.

3. Special Angles

Special angles are specific angles where the values of sine, cosine, and tangent are well-known. These include 0°, 30°, 45°, 60°, and 90°.

Key Values to Memorize

Angle (θ)	sin(θ)	cos(θ)	tan(θ)
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	Undefined

How to Remember Special Angles

• For sine, write the values 0, 1/2, √2/2, √3/2, 1.
• For cosine, write the same values but in reverse.
• For tangent, divide sine by cosine for each angle.

Example Problem

Find tan(45°).

1. Recall that tanθ = sinθ/cosθ.
2. Substitute the values for sin(45°) = √2/2 and cos(45°) = √2/2.
3. Simplify: tan(45°) = (√2/2) / (√2/2) = 1.

Summary of Key Concepts

• Compound Angles: Use specific formulas for sine and cosine of sums and differences.
• Proving Identities: Simplify one side to make it equal to the other using trigonometric rules.
• Special Angles: Memorize sine, cosine, and tangent values for 0°, 30°, 45°, 60°, 90°.

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