Question

If sin(α) =21/29

where 0 < α <π/2
and cos(β) =15/17
where 3π/2
< β < 2π, find the exact values of the following.
(a) sin(α + β)
(b) cos(α − β)
(c) tan(α − β)

Answer 1

Using the sum and difference formulas for sine, cosine, and tangent along with the given values, we can find the exact values for sin(α + β), cos(α - β), and tan(α - β) by also calculating the necessary complementary trigonometric functions.

Step-by-step explanation:

To solve the given trigonometric identities, we use the sum and difference formulas:

• sin (a ± β) = sin a cos β ± cos a sin β
• cos (a ± β) = cos a cos β - sin a sin β
• tan (a ± β) = (tan a ± tan β) / (1 - tan a tan β)

For the first question, given that sin(α) = 21/29 and cos(β) = 15/17, with the angles in specified ranges, we know the following:

• cos(α) can be found using the Pythagorean identity: cos(α) = √(1 - sin^2(α)) = √(1 - (21/29)^2)
• sin(β) is negative in the fourth quadrant, where 3π/2 < β < 2π. It can be found using sin^2(β) + cos^2(β) = 1, thus sin(β) = -√(1 - (15/17)^2)

Now we can calculate sin(α + β) using the formula for sine of sum of angles:

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

Similar approaches will be used for cos(α - β) and tan(α - β) using their respective formulas.

Answer 2

The exact values of sin(α + β) is767/493, cos(α - β) is 493/767, and tan(α - β) and -52/331, we use trigonometric identities and the given values of α and β.

Step-by-step explanation:

In order to find the exact values of the trigonometric functions, we need to use the given information about the angles α and β. Let's start with part (a) sin(α + β).

Using the trigonometric angle addition formula:

sin(α + β) = sin α cos β + cos α sin β.

Substituting the given values:

sin(α + β) = (21/29)(15/17) + (29/21)(17/15) = 767/493.

For part (b) cos(α - β), we use the trigonometric angle subtraction formula:

cos(α - β) = cos α cos β + sin α sin β.
Substituting the given values:

cos(α - β) = (21/29)(15/17) + (29/21)(17/15) = 493/767.

Finally, for part (c) tan(α - β), we can use the identity tan(α - β) = (tan α - tan β)/(1 + tan α tan β).

Substituting the given values:

tan(α - β) = (21/29 - 17/15)/(1 + (21/29)(17/15)) = -52/331.