BackRight Triangles and Evaluating Trigonometric Functions
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Trigonometric Functions and Applications
Right Triangles and Evaluating Trigonometric Functions
This section introduces the definitions and evaluation of trigonometric functions using right triangles, explores special angles, cofunction identities, reference angles, and methods for finding trigonometric values for any angle.
Trigonometric Functions in Right Triangles
Definition: For an acute angle A in standard position, the trigonometric functions are defined as ratios of the sides of a right triangle.
Function | Definition |
|---|---|
Example: Evaluating Trigonometric Functions
Given a right triangle with sides: opposite = 7, adjacent = 24, hypotenuse = 25:
Trigonometric Function Values of Special Angles
Certain angles, such as 30°, 45°, and 60°, have exact trigonometric values that are frequently used in mathematics.
Angle | ||||||
|---|---|---|---|---|---|---|
30° | $2$ | |||||
45° | $1$ | $1$ | ||||
60° | $2$ |
Example: Trigonometric Values for 60°
Cofunction Identities
Cofunction identities relate the trigonometric functions of complementary angles (angles that add up to 90° or radians).
For acute angles and in a right triangle, .
Function | Degree Identity | Radian Identity |
|---|---|---|
Example: Writing Functions in Terms of Cofunctions
Reference Angles
A reference angle for an angle is the positive acute angle formed by the terminal side of $\theta$ and the x-axis. Reference angles are used to find trigonometric values for any angle.
For angles in degrees: Subtract the nearest multiple of 180° or 360° to get an acute angle.
For angles in radians: Subtract the nearest multiple of or as appropriate.
Example: Finding Reference Angles
For , reference angle
For , reference angle
For , reference angle
Finding Trigonometric Function Values for Any Angle
If or , find a coterminal angle by adding or subtracting 360° as needed to bring into .
Find the reference angle .
Find the trigonometric function values for .
Determine the correct sign for the value based on the quadrant in which lies.
Example: Using Reference Angles
: Coterminal with , reference angle . Since is in quadrant II, cosine is negative:
: Coterminal with , reference angle .
Calculator Use and Inverse Trigonometric Functions
To approximate trigonometric values, use a calculator in the correct mode (degree or radian).
Inverse trigonometric functions are used to find angles given a trigonometric value.
Example: If , then
Example: If , then radians
Applications
Trigonometric functions can be used to solve real-world problems, such as finding the angle of a highway grade.
Example: For a 3000-pound car with a grade resistance of 500 pounds,
Finding All Angle Measures Satisfying a Condition
To find all angles in a given interval that satisfy a trigonometric equation, use the reference angle and consider all quadrants where the function has the required sign.
Example: in : Reference angle . Solutions: (quadrants II and III where cosine is negative).
Example: in : ,
Additional info: These notes cover the core Precalculus topics of right triangle trigonometry, special angles, cofunction identities, reference angles, and solving trigonometric equations, as well as calculator use and applications.
