Trig Identities: Angle Addition
There are several trig identities used in mathematics, among which are the angle addition and subtraction formulas. These formulas are summarized as follows:
Angle Addition or Angle Subtraction
The following questions focus on angle addition. Both the identities for sinA+B and cosA+B can be derived using geometry as shown below.
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1. Using the identities in the above table, determine the value of sin(75°).
2. Prove that tanA+B=tanA+tanB1−tanAtanB.
You can use the identity sinA+B=sinAcosB+cosAsinB to determine the value of sin(75°):
First, note that 45° + 30° = 75°. This is important, as these angles form one of the special triangles:
From this, you can substitute the two angles into the identity for sinA+B:
First, note that:
Keeping this in mind, you can substitute the trig identities for sinA+B and cosA+B and get:
Now, divide every term by cosAcosB:
Simplify this expression as follows:
Finally, you can simplify this expression further by recalling that tanQ=sinQcosQ:
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