BackRight Triangle Trigonometry: Trigonometric Functions and Fundamental Identities
Study Guide - Smart Notes
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Trigonometric Functions of Acute Angles
Definition and Overview
Trigonometric functions are defined as ratios of the sides of a right triangle with respect to an acute angle. These functions are fundamental in precalculus and are used to relate angles to side lengths in geometry and trigonometry.
Sine (sin): Ratio of the length of the side opposite the angle to the hypotenuse.
Cosine (cos): Ratio of the length of the side adjacent to the angle to the hypotenuse.
Tangent (tan): Ratio of the length of the side opposite the angle to the side adjacent to the angle.
Cosecant (csc): Reciprocal of sine.
Secant (sec): Reciprocal of cosine.
Cotangent (cot): Reciprocal of tangent.
Function Name | Abbreviation | Value |
|---|---|---|
sine of θ | sin θ | b/c |
cosine of θ | cos θ | a/c |
tangent of θ | tan θ | b/a |
cosecant of θ | csc θ | c/b |
secant of θ | sec θ | c/a |
cotangent of θ | cot θ | a/b |
Example: In a right triangle, if the side opposite θ is 5, the adjacent side is 12, and the hypotenuse is 13, then:
Finding Trigonometric Function Values
Using the Pythagorean Theorem
To find the value of a trigonometric function when only two sides are known, use the Pythagorean Theorem to determine the third side. The theorem states:
Example: Given a right triangle with hypotenuse and adjacent side , find the opposite side :
Evaluating All Six Trigonometric Functions
Once all three sides are known, use the definitions to find each function:
Fundamental Trigonometric Identities
Reciprocal and Quotient Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. The most common are reciprocal and quotient identities:
Pythagorean Identities
Pythagorean identities relate the squares of sine, cosine, tangent, and secant functions:
Finding Values of Remaining Trigonometric Functions
Given One Function Value
If the value of one trigonometric function is known, the values of the others can be found using either triangle definitions or identities.
Option 1: Draw a right triangle, assign side lengths, use the Pythagorean Theorem, and apply definitions.
Option 2: Use identities to solve for the remaining functions.
Example: Given , find the other functions.
Complementary Angles and Cofunctions
Complementary Angle Theorem
Two angles are complementary if their sum is 90°. The cofunctions of complementary angles are equal, which is a fundamental property in trigonometry.
Function | Expression for Angle B | Expression for Angle A |
|---|---|---|
sin B | b/c | cos A |
cos B | a/c | sin A |
tan B | b/a | cot A |
csc B | c/b | sec A |
sec B | c/a | csc A |
cot B | a/b | tan A |
Example: If and are complementary, then .
