It has taken me awhile to try to figure out how to set up today's lesson. It requires a good understanding of TrigValue, which I covered a few weeks ago. This concept appeared a lot--18 times this year, with a median placement at question # 64.
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Trigonometric Identities are equalities that involve certain functions of one or more angles, that are true for every value of the angles. A great list of these identities exists on Wikipedia.
On this year's tests, there were 30 problems related to these identities. 60% of them involved identities of sines and/or cosines, which is why I split them out into this lesson. The following 5 identities were tested this year, in descending order of frequency:
Sine and Cosine Identities
Example 1: 2sin15°cos15° = ___
- Use Identity # 1 above
- sin(2a) = 2sin(a)cos(a) --> sin(2 x 15°) = 2sin15°cos15°
- sin(2 x 15°) = sin(30°) = 1/2
Example 2: If cos 38° = sin B, B ϵ QI, then B = ___°
- B ϵ QI means B is an angle within Quadrant 1 (between 0° and 90°)
- Use Identity # 2 above
- sin(90° - a) = cos(a) --> sin(90° - 38°) = cos(38°)
- sin(52°) = cos(38°), so the answer is 52
Example 6: sin(45°) x cos(135°) = ___
- sin(45°) = 1/√2
- cos(135°) = -1/√2
- (1/√2)(-1/√2) = -1/2
Example 7: Arcsin 1 = πk radians and k = ___
- Arcsin is the inverse function of sin, which means we need to find the angle where sin(a) = 1
- The angle is 90°, which is π/2, or 1/2 (π).
- So our answer is 1/2