**High School Number Sense Lesson 95: Trigonometric Identities--Sine and Cosine**

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**

**1. sin(2a) = 2sin(a)cos(a)**--5 times

**2. sin(90° - a) = cos(a)**--4 times

**3. cos(2a) = cos**--3 times

^{2}(a) - sin^{2}(a)**4. cos(2a) = 2cos**--2 times

^{2}(a) - 1**5. sin**--1 time

^{2}(a) + cos^{2}(a) = 1**TrigValue**that required a bit more mental gymnastics. Let's look at some examples:

**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 3:**

**2cos**

^{2}30° - 2sin^{2}30° = ___1. Use Identity # 3 above

2. Factor out the 2; 2[cos(2 x 30°)] = 2[cos

^{2}30° - sin

^{2}30°]

3. 2cos(60°) = 2(1/2) =

**1**

Example 4:

Example 4:

**2cos**

^{2}(30°) - 1 = ___1. Use Identity # 4 above

2. cos(2 x 30°) = 2cos

^{2}(30°) - 1

3. cos(60°) =

**1/2**

**Example 5:**

**3cos**

^{2}30° + 3sin^{2}30° = ___1. Use Identity # 5 above, and factor out the 3

2. 3(cos

^{2}30° + sin

^{2}30°) = 3(1) =

**3**

**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**

**Example 8:**

**sin**

^{2}390° = ___1. 390° is a full revolution beyond (and equivalent to) 30°

2. sin

^{2}30° = [sin(30°)]

^{2}= (1/2)

^{2}=

**1/4**

**Up Next for High School: Asymptote**