Find the Value Using the Unit Circle sin(15)

Problem

sin(15^∘)

Solution

1. Identify the angle 15^∘ as a difference of two common angles from the unit circle, such as 45^∘ and 30^∘

2. Apply the formula for the sine of a difference, which is sin(α−β)=sin(α)*cos(β)−cos(α)*sin(β)

3. Substitute α=45^∘ and β=30^∘ into the identity.

sin(45^∘−30^∘)=sin(45^∘)*cos(30^∘)−cos(45^∘)*sin(30^∘)

4. Evaluate the trigonometric functions using values from the unit circle.

sin(45^∘)=√(,2)/2

cos(30^∘)=√(,3)/2

cos(45^∘)=√(,2)/2

sin(30^∘)=1/2

5. Multiply the terms together.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

6. Subtract the results to find the final value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin(15^∘)=(√(,6)−√(,2))/4

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