For this case we have the following fraction:
(1-cos ^ 2 (θ)) / (sin ^ 2 (θ))
We must take into account the following trigonometric identity:
cos ^ 2 (θ) + sin ^ 2 (θ) = 1
Therefore rewriting we have:
sin ^ 2 (θ) = 1 - cos ^ 2 (θ)
Substituting in the given fraction we have:
(1-cos ^ 2 (θ)) / (sin ^ 2 (θ))
= (sin ^ 2 (θ)) / (sin ^ 2 (θ))
= 1
Answer:
1
Yes it is a solution, 8-3 is less than or equal to 9
Your percentage of 97.6% as a decimal is 0.976
Answer:
-336/527
Step-by-step explanation:
Trig identities are involved.
tan(2x) = 2tan(x)/(1 -tan(x)²)
Using tan = sin/cos, we can write this in terms of sine and cosine as ...
tan(2x) = (2sin(x)/cos(x))/(1 -sin(x)²/cos(x)²) = 2sin(x)cos(x)/(cos(x)² -sin(x)²)
= 2sin(x)cos(x)/(1 -2sin(x)²)
Now, the cosine can be found from ...
cos(x) = √(1 -sin(x)²)
for sin(θ) = 24/25, cos(θ) = √(1 -(24/25)²) = 7/25 . . . . 1st quadrant angle
Filling in the values in the above identity, we have ...
tan(2θ) = 2(24/25)(7/25)/(1 -2(24/25)²) = 336/-527
tan(2θ) = -336/527
_____
You can use a calculator or estimate that the angle for sin(θ) = 24/25 will be greater than 67.5°, so double the angle will be greater than 135°. (θ ≈ 74°) This means ...
- 2θ > 135°, so the magnitude of tan(2θ) will be less than 1
- 2θ is in the 2nd quadrant, so the sign of tan(2θ) will be negative
These observations will help you choose the correct answer without any further math.
