**Alg 2 BC U12 Days 5 to 8 - Solving Trigonometric Equations and Applications**

**Warm-up: In what QUADRANTS would you look to find:**

**a) angles with a NEGATIVE tangent?**

**b) angles with a POSITIVE cosine?**

**c) angles with a NEGATIVE cosecant?**

**BUT, problems do not always ask for the principal value. Sometimes they ask for other types of answers:**

**Example: Solve for x: **

**a) Give all values of x in ** :

**b) Give all values of x in **:

**c) Give ALL values of x in degrees:**

**d) Give ALL values of x in radians:**

**Using Algebra to solve Trig Equations: (Treat trig functions the same as any other variable.)**

Ex #1: Solve for in the interval :

**Ex #2: Solve for x in the interval **

**Ex #3: Solve for x in the interval **

Problem Set #1:

**Solve for x in the interval** :

1. 2.

**Solve for x in the interval **

3. 4.

**Give ALL solutions for x in radians:**

5. 6.

Give ALL solutions for x in degrees:

7. 8.

Solve for x in the interval .

9. 10.

11. 12.

13. 14.

15. 16. ** Calculator required for 16-24**

Solve for x in (Round to nearest tenth.)

17. 18.

19. 20. (Does not factor … so …)

Solve for x in ** Round to nearest thousandth. **

21. 22. = 0

23. 24.

More equations to try: Page 773: 16-33 and 46-55

Review … to keep you ready for the test!

25. If and is in Quadrant IV, find tan USING IDENTITIES.

26. VERIFY:

Applications:

Angle of Elevation and Angle of Depression: ______formed with the ______!

Example: Shelly knows that when she stands 123 feet from the base of a flagpole, the angle of elevation to the top of the flagpole is 26°40’. If her eyes are 5.3 feet above the ground, find the height of the flagpole. (nearest tenth)

Example: The angle of elevation from the top of a small building to the top of a nearby taller building is 46°40’, while the angle of depression to the bottom of the taller building is 14°10’. If the shorter building is 28 meters high, find the height of the taller building. (nearest tenth)

Example: The length of the shadow of a building 34.09 meters tall is 37.62 meters. Find the angle of elevation of the sun. (nearest tenth of a degree.)

Homework Problems from Book: Page 782: 8, 24, 34, 42, 43, 44, 48

Problem Set #2: 1. The highest mountain peak in the world is Mt. Everest, located in the Himalayas. The height of this enormous mountain was determined in 1856 by surveyors using trigonometry, long before it was first climbed in 1953. This difficult measurement had to be done from a great distance. At an altitude of 14,545 ft on a different mountain, the straight line distance to the peak of Mt. Everest is 27.0134 mi, and its angle of elevation is 5.82°. Approximate the height (to nearest foot) of Mt Everest. (How many feet in a mile?)

2. A researcher is watching a whale approach directly toward a lighthouse as she observes from a telescope at the top of the lighthouse. When she first begins watching the whale, the angle of depression to the whale is 15°50’. Just as the whale turns away from the lighthouse, the angle of depression is 35°40’. If the height of the telescope is 68.7 meters, find the distance traveled by the whale as it approaches the lighthouse, to the nearest tenth of a meter.

3. The length of the shadow of a building 54.03 meters tall is 17.62 meters. Find the angle of elevation of the sun, to the nearest tenth of a degree.

4. If the Empire State Building and the Sears Tower were situated 1000 feet apart, the angle of depression from the top of the Sears Tower to the top of the Empire State building would be 11.53° and the angle of depression from the Top of the Sears Tower to the bottom of the Empire State Building would be 55.48°. Find the heights of the buildings, to the nearest tenth of a foot.

5. From a window 30 ft above the street, the angle of elevation to the top of a taller building across the street is 50°20’, while the angle of depression to the bottom of the taller building is 20°10’. Find the height of the taller building, to the nearest tenth of a foot.

6. From a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 35°, and the angle of elevation to the top of the smokestack is 53°. Find the height of the smokestack, to the nearest tenth of a foot.

7. A swimming pool is 20 meters long. The “floor” of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end. Find the angle of elevation of the “floor” of the pool, to the nearest tenth of a degree.

8. A plane is observed approaching your home and you assume its speed is 600 miles per hour. The angle of elevation to the plane is 16° at one time and 57° one minute later. Approximate the height of the plane, to the nearest tenth of a mile.