sin(x) = 8/17 and 90 degrees < x < 180 degrees then find cos2(x)
Use sin²(x)+cos²(x)=1
and the fact that cos(x) in the second quadrant is negative does not change the value of cos²(x).
To find cos^2(x), we need to first find cos(x) by using the value of sin(x) given to us.
1. Start with the formula for the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
2. Substitute sin(x) = 8/17 into the formula: (8/17)^2 + cos^2(x) = 1.
3. Simplify: 64/289 + cos^2(x) = 1.
4. Subtract 64/289 from both sides: cos^2(x) = 1 - 64/289.
5. Simplify: cos^2(x) = (289 - 64)/289.
6. Further simplify: cos^2(x) = 225/289.
So, cos^2(x) = 225/289, given that sin(x) = 8/17 and 90 degrees < x < 180 degrees.
To find cos^2(x), we need to determine the value of cos(x) and then square it. Here's how you can find the value of cos(x) given that sin(x) = 8/17 and 90 degrees < x < 180 degrees:
1. Recall the relationship between sine and cosine:
- sin(x) = opposite/hypotenuse
- cos(x) = adjacent/hypotenuse
2. Since sin(x) = 8/17, we can assign the following values to the sides of a right triangle:
- opposite = 8 (corresponding to the side opposite to angle x)
- hypotenuse = 17 (corresponding to the hypotenuse of the triangle)
3. To determine the value of the adjacent side, we can use the Pythagorean theorem:
- a^2 + b^2 = c^2, where a and b are the shorter sides, and c is the hypotenuse.
- In our case, a = 8, b is the adjacent side, and c = 17.
Solving for b:
- 8^2 + b^2 = 17^2
- 64 + b^2 = 289
- b^2 = 225
- b = √225 = 15
4. Now we have the values of the adjacent side and the hypotenuse:
- adjacent = 15
- hypotenuse = 17
5. Using the values of the adjacent and hypotenuse sides, we can find cos(x):
- cos(x) = adjacent/hypotenuse
- cos(x) = 15/17
6. Finally, we can square cos(x) to find cos^2(x):
- cos^2(x) = (15/17)^2
You can use a calculator to evaluate cos^2(x) by dividing 15^2 by 17^2, or you can simplify further as follows:
- cos^2(x) = (15/17)^2 = 225/289
So, cos^2(x) is equal to 225/289.