Secondary II Unit 8

Secondary II
Chapter 8 – Trigonometry 2014/2015
Date / Section / Assignment / Concept
A: 11/25 / 8-1 / - Trig Ratios Worksheet / Right Triangle Trigonometry
11/26-28 / Thanksgiving Break
B: 12/1 / 8-1 / - Trig Ratios Worksheet / Right Triangle Trigonometry
A: 12/2
B: 12/3 / 8-2 / - 8-2 Worksheet / Degree Angles, Sketching Angles in Standard Position
A: 12/4
B: 12/5 / 8-3 / - 8-3 Worksheet / Trig Identities
A: 12/8
B: 12/9 / Review
A: 12/10
B: 12/11 / Chapter 8 TEST

Late and absent work will be due on the day of the review (absences must be excused). The review assignment must be turned in on test day. All required work must be complete to get the curve on the test.

Remember, you are still required to take the test on the scheduled day even if you miss the review, so come prepared. If you are absent on test day, you will be required to take the test in class the day you return. You will not receive the curve on the test if you are absent on test day unless you take the test prior to your absence.

Unit 8

8-1 Right Triangle Trigonometry

Reminder- Special Right Triangles

Triangle: Triangle:

Find all missing side lengths.

1. 2.

3. 4.

The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of a right triangle.

C B

Pythagorean Theorem demonstrates a relationship that exists between the sides of a right triangle.

For legs a and b and hypotenuse c:

If is the measure of an acute angle in a right triangle, three trig functions can be defined as followed:

sine: cosine: tangent:

The reciprocals of these functions can also be used to find . They are defined as followed:

cosecant: secant: cotangent:

cscθ=hypotenuseopposite secθ=hypotenuseadjacent cotθ=adjacentopposite

For the following examples, draw and label a right triangle that contains the given information. Find the three trig ratios for the indicated angle.

1. a = 6, b = 8, c = 10, 2. r = 5, s = 12, t = 13,

sin A= sin S=

cos A = cos S =

tan A = tan S =

Calculator Trigonometry Round to 3 decimal places!!!

Use your calculator to find the value. Round to 3 decimal places.

3. sin = 4. tan = 5. cos = 6. cos =

Use your calculator to find the value of the following ANGLES.

Remember: you must use the “2nd” key to find the angle. Round to 3 decimal places.

7. sin x = .500 8. cos B = .7878 9. tan = .1648 10. cos R = .450

x = ______B = ______ = ______R = ______

Find all missing sides and angles for the given right triangles. You may need to use trig ratios (sin, cos or tan).

Sketch a right triangle with the given information.

13. a = 32, b = 15, C = 14. R = , , t = 8

Find A and B and c. Find s, r and

A = ______, s = ______,

B = ______r = ______,

c = ______ = ______

16.

h = ______

17. A boat travels in the following path. How far north did it travel?

18. Rennie is walking her dog. The dog’s leash is 12 feet long and is attached to the dog 10 feet horizontally

from Rennie’s hand, as shown in the diagram. What is the angle formed by the leash and the horizontal at

the dog's collar?

Additional Notes

8-2 Degree Angles, Sketching Angles in Standard Position

An angle in standard position is an angle that has its vertex at the origin, and the initial ray along the positive x-axis. The ending ray is called the terminal ray.

A positive angle in standard position rotates counterclockwise.

A negative angle in standard position rotates clockwise.

For an angle which is drawn in standard position, the reference angle is the positive (acute) angle formed by the terminal side of the angle and the nearest x-axis.

Sketch the following angles in standard position and find the reference angle.

1. 2. 3. 4.

______

Coterminal Angles:

5. Give two angles coterminal with 210º ______

6. Give two angles coterminal with 34º ______

Reminder- Rationalizing the Denominator

When dividing radicals they must be the same “type” root and the terms divide.

If the problem asks you to rationalize your answers you need to simplify so no roots are left in the denominator.

Examples:

a. = = 4 b. = = = (rationalized)

Divide the radicals. Rationalize the denominators.

7. 8. 9.

Special Right Triangles and Trig Ratios

You will frequently need to determine the value of trigonometric ratios for,, and angles on a coordinate system.

Simplest Side Ratios for Special Right Triangles

45°

2 1 2 1

30°

3 45°

Reference Triangles – For each angle measure below, sketch a coordinate system showing all possible angle measures with the given reference angle measure. Then construct the reference triangle for each and label the side lengths. (Hint: The leg lengths may be negative, but the hypotenuse will always be a positive length)

30 degrees 60 degrees 45 degrees

Some angles have a terminal ray which ends on the x or y axis. For these angles we will use the following diagram in calculating trig ratios:

Steps for the following problems:

To evaluate Trig Functions:

1. Sketch the angle in standard position.

2. Calculate the measure of the reference angle. Add this measurement to your diagram.

3. Form a right triangle by extending a segment from the terminal ray perpendicular to the nearest x-axis.

This will form a special right triangle.)

4. Add side lengths to the special triangle. The “legs” may have negative side lengths.

5. Give the exact value of the trig ratios. (Rationalize denominators where necessary).

1. tan = ______2. cos () = ______

3. tan 210˚ = ______4. sin = ______

For the following trig ratios follow the steps above, but use the “Axis” diagram to find the opposite, adjacent, and hypotenuse values.

5. sin = ______6. tan 270˚ = ______

7. If (3, -2) is a point on a graph, give the sin, cos and tan.

Additional Notes

8-3 Trig Identities

Reciprocal Identities

Pythagorean Identities

Prove the following identities:

1. secxcosx=1 2. cscθtanθ=secθ

**REMEMBER**

*Don’t invent new rules.
*Changing things to sin and cos
usually works.
*You can’t use Pythagorean unless things are squared.
*Don’t move things across the equal sign
when proving identities.

4. 5. Simplify.

Additional Notes