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Find the exact values of the six trigonometric functions of the angle θ shown in the
figure.

Find the exact values of the six trigonometric functions of the angle θ shown in the-example-1
User Renat Zamaletdinov
by
3.2k points

1 Answer

23 votes
23 votes

Answer:


\sf \sin(\theta)=(√(2))/(2)


\sf \cos(\theta)=(√(2))/(2)


\sf \tan(\theta)=1


\sf \csc(\theta)=√(2)


\sf \sec(\theta)=√(2)


\sf \cot(\theta)=1

Explanation:


\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}

From inspection of the given right triangle:


  • a = \sf 8

  • c = \sf 8√(2)

Substitute the values of a and c into Pythagoras Theorem and solve for b:


\implies 8^2+b^2=(8√(2))^2


\implies 64+b^2=128


\implies b^2=64


\implies b=8

Therefore, the unknown side length opposite angle θ is 8 units.


\boxed{\begin{minipage}{7 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=(O)/(H)\quad\cos(\theta)=(A)/(H)\quad\tan(\theta)=(O)/(A)$\\\\\\$\sf \csc(\theta)=(H)/(O)\quad\sec(\theta)=(H)/(A)\quad\cot(\theta)=(A)/(O)$\\\\where:\\\phantom{ww}$\bullet$ $\theta$ is the angle\\\phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle\\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle\\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse\\\end{minipage}}

From inspection of the given right triangle:


  • \sf O = 8

  • \sf A = 8

  • \sf H = 8 √(2)

Substitute the values of O, A and H into the trigonometric ratios:


\sf \sin(\theta)=(8)/(8√(2))=(1)/(√(2))=(√(2))/(2)


\sf \cos(\theta)=(8)/(8√(2))=(1)/(√(2))=(√(2))/(2)


\sf \tan(\theta)=(8)/(8)=1


\sf \csc(\theta)=(8√(2))/(8)=√(2)


\sf \sec(\theta)=(8√(2))/(8)=√(2)


\sf \cot(\theta)=(8)/(8)=1

User Simon Michael
by
2.9k points
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