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3 years ago

8

Law of sines: in △fgh, h = 10, m∠f = 65°, and m∠g = 35°. what is the length of g? use the law of sines to find the answer.

2 answers:

gladu [14]3 years ago

5 0

M∠H = 180° - ( 65° + 35° ) = 180° - 100° = 80°
The Law of sines:
h / sin H = g / sin G
10 / sin 80° = g / sin 35°
10 / 0.9848 = g / 0.5736
g = ( 10 · 0.5736 ) : 0.9848 = 5.736 : 0.9848 = 5.8245 ≈ 5.8
Answer: g = 5.8

Margarita [4]3 years ago

4 0

Answer:

I think 5.8 is the correct answer. see picture

Step-by-step explanation:

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See Below.

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We want to verify the equation:

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To start, we can multiply the fraction by (1 - sin(θ)). This yields:

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Simplify. The denominator uses the difference of two squares pattern:

$\displaystyle \frac{1-\sin\theta}{\underbrace{1-\sin^2\theta}_{(a+b)(a-b)=a^2-b^2}} = \sec^2\theta - \sec\theta \tan\theta$

Recall that sin²(θ) + cos²(θ) = 1. Hence, cos²(θ) = 1 - sin²(θ). Substitute:

$\displaystyle \displaystyle \frac{1-\sin\theta}{\cos^2\theta} = \sec^2\theta - \sec\theta \tan\theta$

Split into two separate fractions:

$\displaystyle \frac{1}{\cos^2\theta} -\frac{\sin\theta}{\cos^2\theta} = \sec^2\theta - \sec\theta\tan\theta$

Rewrite the two fractions:

$\displaystyle \left(\frac{1}{\cos\theta}\right)^2-\frac{\sin\theta}{\cos\theta}\cdot \frac{1}{\cos\theta}=\sec^2\theta - \sec\theta \tan\theta$

By definition, 1 / cos(θ) = sec(θ) and sin(θ)/cos(θ) = tan(θ). Hence:

$\displaystyle \sec^2\theta - \sec\theta\tan\theta \stackrel{\checkmark}{=} \sec^2\theta - \sec\theta\tan\theta$

Hence verified.

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