1. Acute Angles and Right Triangles
Trigonometry
Eleventh Edition
Chapter 2
Acute Angles and Right Triangles
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2. 2.1 Trigonometric Functions of Acute Angles

3. Learning Objectives
Right-Triangle-Based Definitions of the Trigonometric Functions
Cofunctions
How Function Values Change as Angle Change
Trigonometric Function Values of Special Angles

4. Right - Triangle - Based Definitions of Trignometric Function
Let A represent any acute angle in standard position.

5. Example 1: Finding Trigonometric Function Values of an Acute Angle (1 of 2)
Find the sine, cosine, and tangent values for angles A and B.

6. Example 1: Finding Trigonometric Function Values of an Acute Angle (2 of 2)
Find the sine, cosine, and tangent values for angles A and B.

7. Cofunction Identities
For any acute angle A, cofunction values of complementary angles are equal.

8. Example 2: Writing Functions in Terms of Cofunction
Write each function in terms of its cofunction.
(a)
(b)
(c)

9. Example 3: Solving Equations Using Cofunctions Identities (1 of 2)
Find one solution for the equation. Assume all angles involved are acute angles.
(a)
Since sine and cosine are cofunctions, the equation is true if the sum of the angles is 90°.
Combine like terms.
Subtract 6°.
Divide by 4.

10. Example 3: Solving Equations Using Cofunctions Identities (2 of 2)
Find one solution for the equation. Assume all angles involved are acute angles.
(b)
Since tangent and cotangent are cofunctions, the equation is true if the sum of the angles is 90°.

11. Increasing/Decreasing Functions
As A increases, y increases and x decreases. Since r is fixed,
Since r is fixed,
sin A increases
csc A decreases
cos A decreases
sec A increases
tan A increases
cot A decreases

12. Example 4: Comparing Function Value of Acute Angles
Determine whether each statement is true or false.
(a) sin 21° > sin 18°
(b) sec 56° ≤ sec 49°
(a) In the interval from 0° to 90°, as the angle increases, so does the sine of the angle, which makes sin 21° > sin 18° a true statement.
(b) For fixed r, increasing an angle from 0° to 90°, causes x to decrease. Therefore, sec θ increases. The given statement, sec 56° ≤ sec 49°, is false.

13. 30°− 60° Triangles (1 of 2)
Bisect one angle of an equilateral triangle to create two 30° − 60° triangles.

14. 30°− 60° Triangles (2 of 2)
Use the Pythagorean theorem to solve for x.

15. Example 5: Finding Trigonometric Function Values for 60° (1 of 2)
Find the six trigonometric function values for a 60° angle.

16. Example 5: Finding Trigonometric Function Values for 60° (2 of 2)
Find the six trigonometric function values for a 60° angle.

17. 45°− 45° Right Triangles (1 of 3)
Use the Pythagorean theorem to solve for r.

18. 45°− 45° Right Triangles (2 of 3)

19. 45°− 45° Right Triangles (3 of 3)

20. Function Values of Special Angles