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Scilla [17]
3 years ago
13

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s). Complete the st

atements below, simplify all ratios and keep them as improper fractions. If cos(θ) = and sin(θ) is negative, then sin(θ) = and tan(θ) = .
Mathematics
1 answer:
Zina [86]3 years ago
5 0

Question:

If cos(θ) =-8/17 and sin(θ) is negative, then sin(θ) = ___  and tan(θ) =___.

Answer:

Sin\theta = \frac{-15}{17}

Tan\theta = \frac{15}{8}

Step-by-step explanation:

Given

cos(θ) =-8/17

Required

sin(θ) = __

tan(θ) =__

The first step is to determine the length of the third side

Given that

cos(\theta) = \frac{Adj}{Hyp}

Where Adj and Hyp represent Adjacent and Hypotenuse

cos(\theta) = \frac{-8}{17}

By comparison

Adj = -8\ and\ Hyp = 17

Using Pythagoras

Hyp^2 = Adj^2 + Opp^2

By Substitution

17^2 = (-8)^2 + Opp^2

289 = 64 + Opp^2

Subtract 64 from both sides

289 - 64 = 64 - 64 + Opp^2

225 = Opp^2

Take square roots of both sides

\sqrt{225} = \sqrt{Opp^2}

\sqrt{225} = Opp

15 = Opp

Opp = 15

The question says that sin(θ) is negative; This implies that θ is in the third quadrant and as such

Opp = -15

From trigonometry

Sin\theta = \frac{Opp}{Hyp}

Sin\theta = \frac{-15}{17}

Also from trigonometry

Tan\theta = Sin\theta / Cos\theta

Tan\theta = \frac{-15}{17} / \frac{-8}{17}

Tan\theta = \frac{-15}{17} * \frac{-17}{8}

Tan\theta = \frac{-15 * -17}{17  * 8}

Tan\theta = \frac{15 * 17}{17  * 8}

Tan\theta = \frac{15}{8}

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