SOLUTION: Evaluate cos(𝛼 + 𝛽) given sin(𝛼) = 5/6, 𝛼 𝑖𝑛 𝑄𝐼𝐼 and tan(𝛽) = 1/2, 𝛽 𝑖𝑛 𝑄𝐼𝐼𝐼

Algebra ->  Trigonometry-basics -> SOLUTION: Evaluate cos(𝛼 + 𝛽) given sin(𝛼) = 5/6, 𝛼 𝑖𝑛 𝑄𝐼𝐼 and tan(𝛽) = 1/2, 𝛽 𝑖𝑛 𝑄𝐼𝐼𝐼      Log On


   

Question 1181851: Evaluate cos(𝛼 + 𝛽) given sin(𝛼) = 5/6, 𝛼 𝑖𝑛 𝑄𝐼𝐼 and tan(𝛽) = 1/2, 𝛽 𝑖𝑛 𝑄𝐼𝐼𝐼
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll use A and B in place of 𝛼 and 𝛽

If sin(A) = 5/6, then cos(A) = -sqrt(11)/6. You can use the pythagorean trig identity cos^2+sin^2 = 1 to see why this works. Or you can draw out a right triangle to find the missing side (using the pythagorean theorem). So it's not a coincidence that the term 'pythagorean' comes up with this trig identity.

Note: cosine is negative in Q2

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If tan(B) = 1/2, then we have an opposite side of 1 and adjacent side of 2.
This leads to a hypotenuse sqrt(1^2+2^2) = sqrt(5), which is again where the pythagorean theorem comes up.

Through a bit of algebra, you should find these two facts:
sin(B) = -sqrt(5)/5
cos(B) = -2sqrt(5)/5

Both sine and cosine are negative in Q3

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In the previous two sections, we found the following four facts
cos(A) = -sqrt(11)/6
cos(B) = -2sqrt(5)/5
sin(A) = 5/6
sin(B) = -sqrt(5)/5

Use those four items in the identity below
cos%28A%2BB%29+=+cos%28A%29%2Acos%28B%29+-+sin%28A%29%2Asin%28B%29



cos%28A%2BB%29+=+2%2Asqrt%2811%2A5%29%2F%286%2A5%29+%2B+5%2Asqrt%285%29%2F%286%2A5%29

cos%28A%2BB%29+=+2%2Asqrt%2855%29%2F30+%2B+5%2Asqrt%285%29%2F30

cos%28A%2BB%29+=+%282%2Asqrt%2855%29+%2B+5%2Asqrt%285%29%29%2F30