evaluate the following in exact form, where the angeles alpha and beta satisfy the conditions:
sin alpha=4/5 for pi/2 < alpha < pi
tan beta=7/24 for pie < beta < 3pi/2
answer choices
A. sin(beta+alpha) B. tan(beta-alpha) C. cos(alpha-beta)
To evaluate the expressions in exact form, we need to find the values of sin(beta+alpha), tan(beta-alpha), and cos(alpha-beta) using the given conditions. Let's solve it step by step:
1. Find the value of alpha:
Given: sin alpha = 4/5 for pi/2 < alpha < pi
To determine the value of alpha, we need to find the angle within the specified range whose sine is 4/5. We can use the inverse sine function (sin^-1) to find the angle:
alpha = sin^-1(4/5)
Using a calculator, we can find the value of alpha to be approximately 0.9273 radians.
2. Find the value of beta:
Given: tan beta = 7/24 for pi < beta < 3pi/2
To determine the value of beta, we need to find the angle within the specified range whose tangent is 7/24. We can use the inverse tangent function (tan^-1) to find the angle:
beta = tan^-1(7/24)
Using a calculator, we can find the value of beta to be approximately 0.2838 radians.
3. Evaluate sin(beta+alpha):
sin(beta+alpha) = sin(beta)cos(alpha) + cos(beta)sin(alpha)
To calculate this, we need to find the values of sin(beta), cos(alpha), cos(beta), and sin(alpha):
sin(beta) = sin(0.2838) (Using the value of beta obtained in Step 2)
cos(alpha) = cos(0.9273) (Using the value of alpha obtained in Step 1)
cos(beta) = cos(0.2838) (Using the value of beta obtained in Step 2)
sin(alpha) = sin(0.9273) (Using the value of alpha obtained in Step 1)
Plug in these values to evaluate sin(beta+alpha) using the formula.
4. Evaluate tan(beta-alpha):
tan(beta-alpha) = (tan(beta) - tan(alpha))/(1 + tan(beta)tan(alpha))
To calculate this, we need to find the values of tan(beta) and tan(alpha):
tan(beta) = tan(0.2838) (Using the value of beta obtained in Step 2)
tan(alpha) = tan(0.9273) (Using the value of alpha obtained in Step 1)
Plug in these values to evaluate tan(beta-alpha) using the formula.
5. Evaluate cos(alpha-beta):
cos(alpha-beta) = cos(alpha)cos(beta) + sin(alpha)sin(beta)
To calculate this, we need to find the values of cos(alpha), cos(beta), sin(alpha), and sin(beta):
cos(alpha) = cos(0.9273) (Using the value of alpha obtained in Step 1)
cos(beta) = cos(0.2838) (Using the value of beta obtained in Step 2)
sin(alpha) = sin(0.9273) (Using the value of alpha obtained in Step 1)
sin(beta) = sin(0.2838) (Using the value of beta obtained in Step 2)
Plug in these values to evaluate cos(alpha-beta) using the formula.
Once you have obtained the numerical values for sin(beta+alpha), tan(beta-alpha), and cos(alpha-beta), you can compare them to the answer choices (A, B, C) and select the correct one.