Trigonometric Identities

You already know the basic trigonometric ratios from SEE. In Grade 11, trigonometry becomes much richer with compound angle formulas, multiple angle formulas, and sum-to-product conversions. These identities are essential tools for simplifying expressions and solving equations.

Fundamental Identities (Review)

sin²A + cos²A = 1

1 + tan²A = sec²A

1 + cot²A = cosec²A

Compound Angle Formulas

These deal with the sine, cosine, and tangent of the sum or difference of two angles:

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

Worked Example

Find the value of sin 75 degrees without a calculator.

Solution:

• sin 75 = sin(45 + 30)
• = sin45 cos30 + cos45 sin30
• = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2)
• = sqrt(6)/4 + sqrt(2)/4
• = (sqrt(6) + sqrt(2)) / 4

Multiple Angle Formulas

Setting B = A in the compound angle formulas:

sin 2A = 2 sinA cosA

cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A

tan 2A = 2tanA / (1 - tan²A)

Half-angle results (from cos 2A formulas):

• cos²A = (1 + cos 2A)/2
• sin²A = (1 - cos 2A)/2

Sum-to-Product Formulas

These convert sums or differences of sines/cosines into products:

sinC + sinD = 2 sin((C+D)/2) cos((C-D)/2)

sinC - sinD = 2 cos((C+D)/2) sin((C-D)/2)

cosC + cosD = 2 cos((C+D)/2) cos((C-D)/2)

cosC - cosD = -2 sin((C+D)/2) sin((C-D)/2)

Product-to-Sum Formulas

2 sinA cosB = sin(A+B) + sin(A-B)

2 cosA cosB = cos(A-B) + cos(A+B)

2 sinA sinB = cos(A-B) - cos(A+B)

Worked Example: Proving an Identity

Prove that: (sin 2A) / (1 + cos 2A) = tan A

Solution (LHS):

• sin 2A / (1 + cos 2A)
• = 2 sinA cosA / (1 + 2cos²A - 1)
• = 2 sinA cosA / (2cos²A)
• = sinA / cosA
• = tan A = RHS

Key Takeaways

• Compound angle formulas let you find exact values of non-standard angles
• Double angle formulas: sin 2A = 2sinAcosA, cos 2A = cos²A - sin²A
• Sum-to-product and product-to-sum formulas convert between forms
• These identities are essential for calculus (integration techniques)