Half Angle Identities
Examples, solutions, videos, worksheets, and activities to help PreCalculus students learn how to derive and use the half angle identities.
The following diagrams show the half-angle identities and double-angle identities. Scroll down the page for more examples and solutions on how to use the half-angle identities and double-angle identities.
Half Angle Identities to Evaluate Trigonometric Expressions
This video gives some half angle identities and show how they can be used to solve some trigonometric equations.
Example:
Find the exact value of the following:
sin(22.5°)
Half Angle Identities to Evaluate Trigonometric Expressions, Example 2.
Example:
Find the exact value of the following:
tan(105°)
Half Angle Identities to Evaluate Trigonometric Expressions, Example 3.
Example:
Find sin(9/2) if cos a = 3/5 for 9° ≤ a ≤ 90°
Solving Trigonometric Equations using Half-Angle Identities
Example:
Solve tan (x/2) + sin x = 0 for x ∊ [0, 2π}
How half-angle identities can be used to determine function values?
Example:
- Determine the exact value of sin(π/8)
- Determine the exact value of cos(105°)
- Given cos A = -2/3 in quadrant II, determine cos(A/2), sin(A/2) and tan(A/2).
Using half-angle formulas in trigonometry
Example:
Let sin A = 4/5 with A in quadrant II.
Find
- sin (A/2)
- cos 2A
- sec 2A
Derive and use Half-Angle Identities
The derivations of the half-angle identities for both sine and cosine, plus listing the tangent ones. Then a couple of examples using the identities.
Examples:
- Find the exact value for sin (9π/8)
- Find cos x/2 if sin x = -4/5 with 3π/2 < x < 2π
How to derive the half angle trigonometry identities for cosine, sine and tangent?
The Half-Angle Identities
The half angle identities come from the power reduction formulas using the key substitution α = θ/2 twice, once on the left and right sides of the equation. With half angle identities, on the left side, this yields (after a square root) cos(θ/2) or sin(θ/2); on the right side cos(2α) becomes cos(θ) because 2(1/2) = 1. For a problem like sin(pi/12), remember that θ/2 = π/12, or θ = π/6, when substituting into the identity.
How to use the power reduction formulas to derive the half-angle formulas?
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