Question

Find exact values of the six trigonometric functions for each angle A.

Accepted Answer

Identify the given angle $$ A $$ and determine if it is in a right triangle context or on the unit circle. This will guide how to find the trigonometric function values. If $$ A  is an $$angle in a right triangle, use the definitions of the six trigonometric functions based on the sides of the triangle: 
- Sine: $$ \sin A = \frac{	ext{opposite}}{	ext{hypotenuse}} 
- $$Cosine: $$ \cos A = \frac{	ext{adjacent}}{	ext{hypotenuse}} 
- $$Tangent: $$ 	an A = \frac{	ext{opposite}}{	ext{adjacent}} 
- $$Cosecant: $$ \csc A = \frac{1}{\sin A} = \frac{	ext{hypotenuse}}{	ext{opposite}} 
- $$Secant: $$ \sec A = \frac{1}{\cos A} = \frac{	ext{hypotenuse}}{	ext{adjacent}} 
- $$Cotangent: $$ \cot A = \frac{1}{	an A} = \frac{	ext{adjacent}}{	ext{opposite}}  If $$the problem provides coordinates or the angle $$ A  is on $$the unit circle, use the coordinates $$ (x, y)  of $$the point on the circle where $$ x = \cos A $$ and $$ y = \sin A . $$Then calculate:
- $$ \sin A = y $$
- $$ \cos A = x $$
- $$ 	an A = \frac{y}{x} $$
- $$ \csc A = \frac{1}{y} $$
- $$ \sec A = \frac{1}{x} $$
- $$ \cot A = \frac{x}{y} $$ Use any given side lengths or coordinates to substitute into the formulas above. Simplify the fractions or expressions to find exact values, often involving radicals or rational numbers. Verify the signs of the trigonometric functions based on the quadrant in which angle $$ A $$ lies, since sine, cosine, and tangent can be positive or negative depending on the quadrant.

Find exact values of the six trigonometric functions for each angle A.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Definition of the Six Trigonometric Functions

Unit Circle and Angle Measurement

Reference Angles and Quadrant Sign Rules