Question

Find the exact value of cosine (alpha beta) given sin alpha = two-fifths for alpha in quadrant ii and cosine beta = one-third for beta in quadrant iv.

Answer 1

Answer:We can use the formula for the cosine of a sum of angles to find cosine (α + β):

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

We are given that sin α = 2/5 and α is in quadrant II, which means that cosine α is negative. Using the Pythagorean identity, we can find cosine α:

cos² α + sin² α = 1

cos² α + (2/5)² = 1

cos² α = 21/25

cos α = -√(21)/5 (since cos α is negative in quadrant II)

We are also given that cosine β = 1/3 and β is in quadrant IV, which means that sine β is negative. Using the Pythagorean identity, we can find sine β:

sin² β + cos² β = 1

sin² β + (1/3)² = 1

sin² β = 8/9

sin β = -2√(2)/3 (since sin β is negative in quadrant IV)

Now we can substitute these values into the formula for cosine (α + β):

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

cos(α + β) = (-√21/5)(1/3) - (2/5)(-2√2/3)

cos(α + β) = -√21/15 + 4√2/15

cos(α + β) = (-√21 + 4√2)/15

Therefore, the exact value of cosine (α + β) is (-√21 + 4√2)/15.

Step-by-step explanation: