Trig Identities

Sum and Difference Identities

Use the sine addition formula to compute an exact value for a non-standard angle by writing it as a sum of two special angles.

To find sin(75°): write 75° = 45° + 30°, substitute the exact special-angle values (√2/2, √3/2, 1/2), multiply, and combine radicals over the common denominator.

sum-difference The sum formula for sine: sin(α+β) = sin α cos β + cos α sin β

Example

Use the sine sum identity to compute an exact nonstandard angle value.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

sin⁡(75∘)=sin⁡(45∘+30∘)\sin( 75 ^\circ)=\sin( 45 ^\circ+ 30 ^\circ)

Step 2 — Sum formula

Use the sine sum identity.

sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin(a+b)=\sin a\cos b+\cos a\sin b

Step 3 — Substitute exact values

Substitute the exact special-angle values.

22⋅32+22⋅12\hlmath{\frac{\sqrt{2}}{2}} \cdot \hlmath{\frac{\sqrt{3}}{2}} + \hlmath{\frac{\sqrt{2}}{2}} \cdot \hlmath{\frac{1}{2}}

Step 4 — Multiply terms

Multiply to get root 6 over 4 plus root 2 over 4.

64+24\hlmath{\frac{\sqrt{6}}{4}} + \hlmath{\frac{\sqrt{2}}{4}}

Step 5 — Result

Combine the terms for sin 75 degrees.

sin⁡(75∘)=6+24\sin( 75 ^\circ)= \hlmath{\frac{\sqrt{6}+\sqrt{2}}{4}}