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 Q: the side opposite $$ 	heta  is$$ $$ 3 $$, the side adjacent to $$ 	heta  is$$ $$ 4 $$, and the hypotenuse $$ PQ  is $$unknown. Use the Pythagorean Theorem to find the hypotenuse $$ PQ $$: $$ PQ^2 = QR^2 + PR^2 $$, which translates to $$ PQ^2 = 4^2 + 3^2 . $$Calculate $$ PQ  by $$taking the square root of the sum: $$ PQ = \sqrt{4^2 + 3^2} . $$This gives the length of the hypotenuse. Find the six trigonometric functions of $$ 	heta $$ using the side lengths: 
- Sine: $$ \sin 	heta = \frac{	ext{opposite}}{	ext{hypotenuse}} = \frac{3}{PQ} 
- $$Cosine: $$ \cos 	heta = \frac{	ext{adjacent}}{	ext{hypotenuse}} = \frac{4}{PQ} 
- $$Tangent: $$ 	an 	heta = \frac{	ext{opposite}}{	ext{adjacent}} = \frac{3}{4} $$ Use the reciprocal identities to find the remaining three functions:
- Cosecant: $$ \csc 	heta = \frac{1}{\sin 	heta} = \frac{PQ}{3} 
- $$Secant: $$ \sec 	heta = \frac{1}{\cos 	heta} = \frac{PQ}{4} 
- $$Cotangent: $$ \cot 	heta = \frac{1}{	an 	heta} = \frac{4}{3} $$

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.

Verified video answer for a similar problem:

Key Concepts

Pythagorean Theorem

Right Triangle Trigonometric Functions

Identifying Sides Relative to an Angle