Apply Pythagorean identities to simplify and verify expressions. Math 30-1.
Lesson 5.3 of Trigonometric Identities in Math 30-1 — Alberta curriculum lessons.
For any two angles α and β, we have the following sum and difference identities. All four are on your formula sheet.
Sine Identities. (α + β) = αβ + αβ
(α - β) = αβ - αβ
Notice: the sign in the result matches the sign in the angle.
Cosine Identities. (α + β) = αβ - αβ
(α - β) = αβ + αβ
Notice: the sign in the result is opposite to the sign in the angle.
The sum and difference identities can be used to find the exact values of trig ratios. The key idea is to break an angle into two angles whose exact trig values you already know — like 30°, 45°, 60°, 90°, etc.
When you see a trig expression, always look to see if it matches the pattern of one of the sum or difference identities — if it does, you can collapse it into a single ratio immediately.
Sometimes you won't be given the angle directly. Instead you'll be given trig ratio values and the quadrant, and you'll need to find the missing side lengths using the Pythagorean theorem before applying the identity.
Find the exact value of 15°. 15°
Break 15° into two known angles: 15° = 45° - 30°
Step 1 — Apply the sine difference identity. 15° = (45° - 30°)
= 45° 30° - 45° 30°
Step 2 — Substitute exact values. Use the known exact values:
45° = 45° = (√(2))/(2), 30° = (√(3))/(2), 30° = (1)/(2)
= (√(2))/(2) · (√(3))/(2) - (√(2))/(2) · (1)/(2)
= (√(6))/(4) - (√(2))/(4)
= (√(6) - √(2))/(4)
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