To answer this question, we can graph the situation as follows:
We graphed a right triangle to express one of the given trigonometric ratios: sin(Θ). We need to find x using the Pythagorean Theorem to find the values of the other five trigonometric functions. Then we have:
[tex]x^2+84^2=85^2\Rightarrow x^2=85^2-84^2\Rightarrow\sqrt[]{x^2}^{}=\sqrt[]{85^2-84^2}[/tex]
Then, we have:
[tex]x=\sqrt[]{169}\Rightarrow x=13[/tex]
Now we can express the exact values of the five remaining trigonometric functions:
The exact value for cosine function:
[tex]\cos (\theta)=\frac{\text{adj}}{\text{hyp}}\Rightarrow\cos (\theta)=\frac{13}{85}[/tex][tex]\tan (\theta)=\frac{opposite}{\text{adjacent}}=\frac{84}{13}\Rightarrow\tan (\theta)=\frac{84}{13}[/tex]
The exact value for the tangent function is shown above.
Now the other exact values for the other trigonometric functions are:
Cosecant:
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