If sin (theta+24)=costheta where theta is acute angle .find theta
since cos(x) = sin(90-x)
sin(x+24) = sin(90-x)
x+24 = 90-x
2x = 66
x = 33
check:
sin(57) = cos(33)
yep
To find the value of theta in the equation sin(theta+24) = cos(theta), we can use some trigonometric identities and properties.
1. First, let's simplify the equation using trigonometric identities.
sin(theta+24) = cos(theta)
sin(theta)cos(24) + cos(theta)sin(24) = cos(theta)
sin(theta)cos(24) = cos(theta) - cos(theta)sin(24)
sin(theta)cos(24) = cos(theta)(1 - sin(24))
2. Now, we can use the fact that theta is an acute angle, which means its range is between 0 and 90 degrees.
Since theta is an acute angle, both sin(theta) and cos(theta) will be positive.
Since both sin(theta) and cos(theta) are positive, we can cancel them from both sides of the equation.
cos(24) = 1 - sin(24)
3. Next, we can solve for sin(24) using a scientific calculator.
cos(24) ≈ 0.9135 (rounded to four decimal places)
4. Substitute the value of cos(24) into the equation.
0.9135 = 1 - sin(24)
5. Solve for sin(24).
sin(24) = 1 - 0.9135
sin(24) ≈ 0.0865 (rounded to four decimal places)
6. Now, we can find the value of theta using the inverse sine function (sin^(-1)).
theta = sin^(-1)(0.0865)
theta ≈ 4.9779 degrees (rounded to four decimal places)
Therefore, the value of theta, in degrees, that satisfies the equation sin(theta+24) = cos(theta) is approximately 4.9779 degrees.