Question

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

Accepted Answer

Identify the sides of the right triangle relative to angle $$ 	heta  at $$vertex R. The hypotenuse is the side opposite the right angle, which is $$ PR = 53 . $$The side opposite $$ 	heta  is$$ $$ PQ = 28 $$, and the adjacent side to $$ 	heta  is$$ $$ QR $$, which is unknown. Use the Pythagorean Theorem to find the missing side $$ QR . $$The theorem states:

$$QR^2$$ + $$PQ^2$$ = $$PR^2$$

Substitute the known values:

$$QR^2$$ + $$28^2$$ = $$53^2$$ Solve for $$ QR^2 $$:

$$QR^2$$ = $$53^2$$ - $$28^2$$

Then take the square root to find $$ QR $$:

$$QR = \sqrt{53^2$ - 28^2$}$$ Once you have the length of $$ QR $$, find the six trigonometric functions of $$ 	heta $$ using the definitions relative to angle $$ 	heta $$:

- $$ \sin 	heta = \frac{	ext{opposite}}{	ext{hypotenuse}} = \frac{PQ}{PR} $$  
- $$ \cos 	heta = \frac{	ext{adjacent}}{	ext{hypotenuse}} = \frac{QR}{PR} $$  
- $$ 	an 	heta = \frac{	ext{opposite}}{	ext{adjacent}} = \frac{PQ}{QR} $$  
- $$ \csc 	heta = \frac{1}{\sin 	heta} = \frac{PR}{PQ} $$  
- $$ \sec 	heta = \frac{1}{\cos 	heta} = \frac{PR}{QR} $$  
- $$ \cot 	heta = \frac{1}{	an 	heta} = \frac{QR}{PQ} $$ Substitute the known side lengths into these formulas to express each trigonometric function in terms of numbers. This completes the process of finding the missing side and the six trigonometric functions of $$ 	heta .$$

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

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Concetti chiave

Pythagorean Theorem

Right Triangle Trigonometric Functions

Identifying Sides Relative to an Angle