College Trigonometry with Solid Mensuration

College Trigonometry with Solid Mensuration

1. MATH 166 College Trigonometry with Solid Mensuration Trigonometry 11e by R. Aufmann

2. Angles and Angular Measure Angles formed by a counterclockwise rotation are considered positive angles , and angles formed by a clockwise rotation are considered negative angles .

3. Degree Measure hich can be written a The measure of an angle is determined by the amount of rotation of the initial side. An angle formed by rotating the initial side counterclockwise exactly once until it coincides with itself (one complete revolution) is defined to have a measure of 360 degrees, w s 360°. = of a revolution

4. Angle Measures Angles are often classified according to their measure. ➢ ➢ ➢ ➢ 180 o angles are straight angles . 90 o angles are right angles . Angles that have a measure greater than o but less than 90 o are acute angles . Angles that have a measure greater than 90 o but less than 180 o are obtuse angles .

5. Angle Measures An angle superimposed in a Cartesian coordinate system is in standard position if its vertex is at the origin and its initial side is on the positive x- axis. Two positive angles are complementary angles if the sum of the measures of the angles is 90 o . Each angle is the complement of the other angle. Two positive angles are supplementary angles if the sum of the measures of the angles is 180 o . Each angle is the supplement of the other angle.

6. Radian Measure Another commonly used angle measurement is the radian . To define a radian, first consider a circle of radius r and two radii and . The angle formed by the two radii is a central angle . The portion of the circle between and is an arc of the circle. We say that arc subtends the angle . The length of arc is . One radian is the measure of the central angle subtended by an arc of length on a circle of radius .

7. Conversion Examples: Convert each angle in degrees to radians. 60 o b) 315 o c) - 150 o Convert each angle in radians to degrees. 1 radian

8. s= r ⋅ θ

9. What is the complementary angle to 37° ? An angle measures 112° . What is its supplementary angle? Two angles are supplementary . If one angle is 4 times the measure of the other, what are the measures of the two angles? If angle A and angle B are complementary , and the measure of angle A is (2x + 10)° and the measure of angle B is (x + 20)° , what is the value of x ? Two angles form a linear pair . If one angle measures 85° , what is the measure of the other angle, and are these angles complementary or supplementary?

10. Special Angles

11. Special Angles

12. CO1 – Lesson 1.2 Trigonometric Functions

13. Introduction Trigonometry Triangle Measurement When working with right triangles, it is convenient to refer to the side opposite an angle or the side adjacent to (next to) an angle.

14. The Six Trigonometric Functions Six ratios can be formed by using two lengths of the three sides of a right triangle. Each ratio defines a value of a trigonometric function of a given acute angle . The functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

15. Fundamental Identities

16. Unit circle

17. Special Angles

18. Unit circle

19. asdsads

20. Sun and Difference Identities

21. Exercises

22. Introduction Some applications require that a product of trigonometric functions be written as a sum or difference of these functions. Other applications require that the sum or difference of trigonometric functions be represented as a product of these functions. The product-to-sum identities are particularly useful in these types of problems.

23. Product-to-Sum Identities The 4 Sum-to-Product identities have corresponding Product-to-Sum identities:

24. Examples Find the exact value of . Solution: We use the identity with and .  

25. Examples Write the difference as a product. Solution:  

26. Exercise Find the exact value of cos 165° - cos 75°. Write the sum as a product.  

27. Exercise Find the exact value of the following: a) b) 2. Write the following product as a sum: a) b) 3. Write the following sum as product: a) b) 4. Verify the following identities: a) b)  

28. DOUBLE – ANGLE IDENTITIES

29. HALF – ANGLE IDENTITIES

30. APPLICATIONS Angle of elevation/ depression Angle of elevation Angle of depression

31. APPLICATIONS Bearing of a line N S E W     What are the bearing of the two lines?

32. ILLUSTRATIONS: From the top of the cliff which rises vertically 168.5 ft above the river bank, the angle of depression of the opposite bank is . How wide is the river? A man standing 230 ft from the foot of building finds that the angle of elevation of the top of the building is . If his eye is above the ground, what is the height of the building?  

33. ILLUSTRATIONS: A surveyor found the angle of elevation of the top of a building to be . After walking towards the building, the angle of elevation measured is . How tall is the building if the device used for surveying has a height of ?  

34. ILLUSTRATIONS: From a certain point, a ship sails 45 miles due north and then proceeds westward at a speed of 20 mph. Find the bearing of the ship and its distance from the starting point four hours after it turned westward? A ship sails and travels at a bearing of for two hours before changing its bearing to for additional 3 hours. If the ship travels at a constant speed of , how far is the ship from its original location?  

35. EXERCISES: From the top of a lighthouse above sea level, a boat is observed under an angle of depression of . How far is the boat from the lighthouse?   Two strings tether a balloon on the ground, as shown. How high is the balloon?     10 m From a certain point, a boat sails 30 miles due south and then proceeds eastward at a speed of 15 mph. Find the bearing of the boat and its distance from the starting point 5 hours after it turned eastward.