Trig identities

Question

Find sin y if sec y= square root of 30 divided by 5 and tan y<0

Answer 1

Let's first determine which quadrant our angle is in. We know that the tangent needs to be negative and tangent is negative in Quadrant II & IV. We also know that our angle has to have a positive cosine and cosine is positive in Quadrant I & IV. Therefore, it looks like both conditions are met in Quadrant IV and sine in QIV is negative.

Now let's look at what else we are given. We have sec y = √(30)/5. We know that sec is also equal to 1/cos--->

cos=1/sec. This gives us cos y = 5/√(30). Remembering that cosine = adjacent/hypertenuse, this tells us that the adjacent = 5 and the hyp. = √(30). Knowing that the hyp.² = adj.² + opp.² we can find the opp. side:

hyp.² = adj.² + opp.²

(√(30))² = 5² + opp.²

30 = 25 + opp.²

5 = opp.²

opp. = ±√5

sin = opp./hyp.

Sine is negative in QIV, so opp. = -√5. This gives sin = opp./hyp.--->-√5/√30--->-√(1/6). Therefore, sin y = -√(1/6)--->-√(6)/6.

Answer 2

sec(y) = 1/cos(y) = √30/5 so cos(y) = 5/√30 = √30/6

sin²(y) + cos²(y) = 1

sin²(y) = 1 - cos²(y)

sin²(y) = 1 - 30/36

sin(y) = ±√(6/36) = ±√(1/6) = ±√6/6

Tan(y)>0 in quadrants II and IV. Sec(y)>0 in quadrants III and IV. Therefore sin(y) is in quadrant IV, where it is negative; so sin(y) = -√6/6

Answer 3

Draw a figure.

sin(y) = (1/5)√5/[(1/5)√30 = sqrt(5/30) = sqrt(1/6) = .408428

and where tan y < 0 then y = π-.420534 = 2.721059