## Unit 7: Advanced Trigonometry

**IOWA CORE STANDARDS ADDRESSED IN THIS UNIT**

- Apply trigonometry to general triangles
- Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles
- Extend the domain of trigonometric functions using the unit circle.
- Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
- Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
- Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for
*x*, π +*x*, and 2π –*x*in terms of their values for*x*, where*x*is any real number.

**Part 1: Law of SInes and Cosines**

DECIDING which method is best to use: Law of Sines or Law of Cosines

If you are given SAS or SSS, you should use Law of Cosines. When you are given anything else, you should use Law of Sines. Before you begin the IXL assignment, watch this video to help you understand the differences in problems and how to approach them.

Also, remember, you should not use an estimated answer to find another value. Always use given values in any formula.

If you are given SAS or SSS, you should use Law of Cosines. When you are given anything else, you should use Law of Sines. Before you begin the IXL assignment, watch this video to help you understand the differences in problems and how to approach them.

Also, remember, you should not use an estimated answer to find another value. Always use given values in any formula.

**Assignment 1**: IXL Y.19: Solving a triangle using Law of Cosines or Sines. Must reach a smart score of 100 to show mastery.

**Part 2: Deciding Law of Cosines, Law of Sines or SOH CAH TOA in application problems.**

**Assignment 2**: Applications of Law of Cosines, Law of Sines and SOH CAH TOA. In the following problems, you will sketch a picture based on the information given, decide which method you should use to solve the problem, and answer the question.

Law of Cosines and Law of Sines

1. A circle is circumscribed about a regular octagon with a perimeterof 48 cm. Find the diameter of the circle.

2. A ship’s captain plans to sail to a port. He first planned to sail from his starting point to the port that is 450 miles at a 12∞ angle

east of north. Due to a hurricane, he starts out sailing straight north, then needs to turn 17∞ east of north to reach the port. How

far does he need to travel in this direction to reach port?

3. A corner of McCormick Park occupies a triangular area that faces two streets that meet at an angle measuring 85∞. The sides of the area facing the streets are each 60 feet in length. The park’s landscaper wants to plant begonias around the edges of the

triangular area. Find the perimeter of the triangular area.

4. The sides of a triangle measure 6.8, 8.4 and 4.9 cm. Find the measure of the smallest angle.

5. A parallelogram has sides of 55 cm and 71 cm. Find the length of each diagonal if the largest angle of the parallelogram measures 106.

6. Two ships leave San Francisco at the same time. In the next 11 hours, one travels 40∞ west of north for 220 miles. The other

travels 10∞ west of south for 165 miles. How far apart are they now?

7. Two surveyors 560 yards apart along the edge of a canyon sight a boundary marker C between them on the other side of a canyon at angles of 27 and 38. Their measurements will be used to plan a bridge that perpendicularly spans the canyon. How long will the bridge need to be?

8. Annie and Sashi are backpacking in the Sierra Nevada. They walk 8 km from their base camp at a bearing of 42∞. After lunch, they change direction to a bearing of 137∞ and walk another 5 km. How far are they now from their base camp? At what bearing mush Sashi and Annie travel to return to their base camp?

## Advanced Trigonometry

**TOPIC ONE: Angles and their measures**

**TOPIC TWO: Central Angles and Arcs**

**Examples: Arc Lengths and Areas**

Find the arc length and area of a sector with central angle 23 and radius 6 cm.

Find the arc length and area of a sector with central angle 24 and diameter of 21m.

The arc length of a sector is 61.3. If the central angle is 270 , what is the radius and area of the sector?

The area of a sector is 160cm2. Find the central angle and arc length if the radius of the circle is 30.