Answer and Explanation:
We are being asked to find the trigonometric ratios for various angles in triangle LMN.
The trigonometric ratios are defined as:
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
They can be remembered using the mnemonic:
SOH-CAH-TOA
1) sin(L) = opp / hyp
= 6/10
= 3/5
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2) tan(N) = opp / adj
= opp / 6
We can solve for the opposite side length using the Pythagorean Theorem:
a² + b² = c²
opp² + 6² = 10²
opp² = 10² - 6²
opp = [tex]\sqrt{10^2 - 6^2}[/tex]
opp = [tex]\sqrt{100-36}[/tex]
opp = [tex]\sqrt{64}[/tex]
opp = 8
Now, we can finish the tangent ratio:
tan(N) = 8 / 6
= 4/3
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3) sin(N) = opp / hyp
= 8 / 10
= 4/5
4) cos(L) = adj / hyp
= 8 / 10
= 4/5
