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Exact Values using identities

Q: Find the exact values:

(a)  sin(5π/12)

(b)  cos(5π/12)

(c)  tan(5π/12)

A:  In order to solve this, I am going to convert it to degrees to start (you don’t need to do this, but I just feel like it)…

Convert like so:  5π/12 * 180/π = 75°

(a)

Now, there are many different identities you can use to solve this, but I am going to solve it like so:

sin(5π/12) = sin(75°) = sin(30° + 45°)

Now we can use the angle sum identity that says:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

sin(30° + 45°) = sin(30°)cos(45°) + cos(30°)sin(45°)

All of the values on the right side of the equation are exact values that you should have memorized (or that you can look up):

sin(30° + 45°) = (1/2)(√(2)/2) + (√(3)/2)(√(2)/2)

sin(30° + 45°) = √(2)/4 + √(6)/4 = [√(2) + √(6)] / 4

(b)  Same concept for this part!

cos(5π/12) = cos(75°) = cos(30° + 45°)

And, the identity says:

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

cos(30° + 45°) = cos(30°)cos(45°) – sin(30°)sin(45°)

Again, the values on the right side are ones you should know or look up!

cos(30° + 45°) = (√(3)/2)(√(2)/2) – (1/2)(√(2)/2)

cos(30° + 45°) = √(6)/4 – √(2)/4 = [√(6) – √(2)]/4

(c)  Same thing again for this part!

tan(5π/12) = tan(75°) = tan(30° + 45°)

And, the identity says:

tan(a + b) = [tan(a) + tan(b)] / [1 – tan(a)tan(b)]

tan(30° + 45°) = [tan(30°) + tan(45°)] / [1 – tan(30°)tan(45°)]

tan(30° + 45°) = [tan(30°) + tan(45°)] / [1 – tan(30°)tan(45°)]

tan(30° + 45°) = [√(3)/3 + 1] / [1 – √(3)/3*1]

tan(30° + 45°) = [√(3)/3 + 1] / [1 – √(3)/3]

If you rationalize the denominator and simplify, you get:

tan(30° + 45°) = √(3) + 2

One thought on “Exact Values using identities

  1. for the cos answer , be written as -square root over 2/4 (square root 3+1)?

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