Trigonometric Functions of Acute Angles

Definition and Right Triangle Relationships

Trigonometric functions are defined as ratios of the sides of a right triangle relative to an acute angle. These functions are fundamental in precalculus and are used to relate angles to side lengths in geometry and applications.

• Sine (sin θ): Ratio of the length of the side opposite θ to the hypotenuse.

• Cosine (cos θ): Ratio of the length of the side adjacent to θ to the hypotenuse.

• Tangent (tan θ): Ratio of the length of the side opposite θ to the side adjacent to θ.

• Cosecant (csc θ): Reciprocal of sine; hypotenuse divided by opposite.

• Secant (sec θ): Reciprocal of cosine; hypotenuse divided by adjacent.

• Cotangent (cot θ): Reciprocal of tangent; adjacent divided by opposite.

Example: Finding Trigonometric Function Values

Given a right triangle with sides labeled, the values of the six trigonometric functions can be determined using the ratios above.

To find the missing side, use the Pythagorean Theorem:

• For example, if , , then

Now, the trigonometric functions are:

Fundamental Trigonometric Identities

Reciprocal and Quotient Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. The most basic are the reciprocal and quotient identities:

• Reciprocal Identities:

• Quotient Identities:

Pythagorean Identities

Pythagorean identities relate the squares of sine and cosine to 1, and are derived from the Pythagorean Theorem:

Finding Values of Remaining Trigonometric Functions

Given One Function Value

If the value of one trigonometric function is known for an acute angle, the values of the other five can be found using either triangle definitions or identities.

• Option 1: Using Triangle Definitions

  1. Draw a right triangle with the given function value.

  2. Assign values to two sides based on the function.

  3. Find the third side using the Pythagorean Theorem.

  4. Calculate the remaining functions using ratios.

• Option 2: Using Identities

  1. Use Pythagorean and reciprocal identities to find other function values.

Example: Given sin θ, Find All Functions

Suppose and θ is acute. Draw a triangle with opposite side and hypotenuse . Find the adjacent side:

Now, calculate:

Example: Given tan θ, Find All Functions

Suppose and θ is acute. Draw a triangle with adjacent side and opposite side . Find the hypotenuse:

Now, calculate:

Complementary Angle Theorem and Cofunction Identities

Theorem and Identities

The complementary angle theorem states that the cofunction of an angle is equal to the function of its complement. For acute angles A and B in a right triangle, where :

Example: Using Cofunction Identities

Find the exact value of expressions involving complementary angles, such as and :

Summary Table: Trigonometric Functions and Their Relationships

Additional info: All examples and identities are fundamental to the study of trigonometric functions in precalculus, and are essential for solving problems involving right triangles, angles, and their relationships.