Use the angle difference identity to find the exact value of each. 3) cos 75 0 4) sin 105 0 5) show sin 4θ = 4sin θ cos θ cos 2 θ 6) solve 5 cos 2 θ = 1 for 0 o < θ < 360 o 7) solve cos 2 θ = cos θ for 0 o < θ < 360 o . 8) solve cos 2 θ = cos θ + 2 for 0 < θ < 2p

3) cos 75 = (√6 - √4)/4  

4) sin 105 = (√6 + √4)/4  

5) below

6) θ = 39.2° or 140.7°

7) θ = 120° or 240°

8) θ = 0.267π rad or π rad

Step-by-step explanation:

3) cos 75 = cos (30+45) = cos30.cos45 - sin30sin45=

√3/2 . √2/2 - 1/2 . √2/2 = (√6 - √4)/4

4)sin 105 = sin (60+45) = sin60.cos45 + sin45.cos60 =

√3/2 . √2/2 + 1/2 . √2/2 = (√6 + √4)/4

5)sin4θ = 4sinθcosθcos2θ

sin4θ = sin(2θ+2θ) = sin2θcos2θ + sin2θcos2θ = 2sin2θcos2θ

As sin2θ = sinθcosθ + sinθcosθ = 2sinθcosθ

substituting:

2.2sinθcosθ.cos2θ = 4sinθcosθcos2θ

6) 5cos2θ = 1               0°<θ<360°

cos2θ = 1/5

As cos2θ = cosθcosθ - sinθsinθ = cos²θ - sin²θ

and sin²θ + cos²θ = 1 → sin²θ = 1- cos²θ

we can say that cos2θ = cos²θ - (1- cos²θ) = cos²θ - 1 + cos²θ = 2cos²θ -1

So, 2cos²θ - 1 = 1/5

2cos²θ = 1/5 + 1

2cos²θ = 6/5

cos²θ = 6/10

cos²θ = 3/5

cosθ = ± √(3/5)

θ = +cos⁻¹√(3/5)  → θ = 39.2°

θ = -cos⁻¹√(3/5)  → θ = 140.7°

7) cos2θ = cosθ                    0°<θ<360°

From the item above, we know that cos2θ = 2cos²θ - 1

2cos²θ - 1 = cosθ

2cos²θ - cosθ - 1 = 0

Making cosθ = y to facilitate:

2y² - y - 1 = 0

Δ = (-1)² - 4.2.(-1) = 9

√Δ = 3

y = (1±3)/4

y₁ = 4/4 = 1

y₂ = -2/4 = -1/2

cosθ = y

cosθ = 1 → θ = 0°

cosθ = -1/2 → θ = 120° or 240°

As  0°<θ<360° (no equal sign) → θ = 120° or 240°

8) cos2θ = cosθ + 2 for 0 < θ < 2p

From the item above, we know that cos2θ = 2cos²θ - 1

2cos²θ - 1 = cosθ + 2

2cos²θ - cosθ - 3 = 0

Making cosθ = y to facilitate:

2y² - y - 3 = 0

Δ = (-1)² - 4.2.(-3) = 25

√Δ = 5

y = (1±5)/4

y₁ = 6/4 = 2/3

y₂ = -4/4 = -1

cosθ = y

cosθ = 2/3 → θ = 48.2° = 0.267π rad

cosθ = -1 → θ = 180° = π rad

this is not the answer this is just a question. is this middle school math?

step-by-step explanation:

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: