If cos α = 0.916 and sin β = 0.122 with both angles’ terminal rays in Quadrant-I, find the values of tan ( α + β ) =

Answer 1

Like the previous question I answered, the identity sin²x + cos²x = 1 will be useful.

cos² x = 1 - sin²x, which means cos x = √1 - sin²x

sin² x = 1 - cos² x, which means sin x = √1 - cos²x

In this case we know cos a = 0.916. This means that sin a, according to our earlier formula is √1 - cos²a = √1 - 0.916² = √0.160944 ≈ 0.401

Since tan a = sin a / cos a, tan a ≈ 0.401/0.916 ≈ 0.438

We also know that sin b = 0.122. This means that cos b, according to our earlier formula is √1 - sin²a = √1 -0.122² = √0.985 ≈ 0.993

This means that tan b = sin b/cos b ≈ 0.122/ 0.993 ≈ 0.123

Finally, we can use our addition formula for tan. tan (a+b) = tan a + tan b / (1 - tan a tan b)

In this case, tan (a+b) = (0.438 + 0.123)/(1 - 0.438*0.123) = 0.561/(1-0.053874) = 0.561/0.946 ≈ 0.593.