If x is between 0 and 90 what is sin x if sinx^(1/2)=cosx. How can I do it without a calculator??
square both sides
write cos^2 = 1-sin^2
Then you have a quadratic in sinx
cos2=0.1-sin2=1square bothside you ll get your answer
To find the value of sin x when sin(x)^(1/2) = cos(x), let's go through the steps without using a calculator:
1. Start by using the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Since sin(x)^(1/2) = cos(x), we can rewrite it as sin(x) = cos^2(x).
2. Substitute cos^2(x) for sin(x) in the Pythagorean identity: cos^2(x) + cos^2(x) = 1.
3. Simplify the equation: 2 cos^2(x) = 1.
4. Divide both sides by 2: cos^2(x) = 1/2.
5. Take the square root of both sides to solve for cos(x): cos(x) = ±sqrt(1/2).
6. Since x is between 0 and 90 degrees, we can determine the quadrant by analyzing the signs of sin(x) and cos(x) in that range.
7. In the first quadrant (0-90 degrees), both sin(x) and cos(x) are positive. Therefore, cos(x) = sqrt(1/2).
8. Now that we have the value of cos(x), we can use the Pythagorean identity once again to find sin(x): sin^2(x) = 1 - cos^2(x).
9. Substitute the value of cos(x): sin^2(x) = 1 - (sqrt(1/2))^2.
10. Simplify: sin^2(x) = 1 - 1/2.
11. Further simplify: sin^2(x) = 1/2.
12. Take the square root of both sides: sin(x) = ±sqrt(1/2).
13. Since x is between 0 and 90 degrees, both sin(x) and cos(x) are positive. Thus, sin(x) = sqrt(1/2).
Therefore, when x is between 0 and 90 degrees, if sin(x)^(1/2) = cos(x), then sin(x) = sqrt(1/2).