Download Secondary II

Secondary II
Unit 8
Secondary II
Chapter 8 – Trigonometry
Date
A: 11/25
Section Assignment
8-1
2014/2015
Concept
- Trig Ratios Worksheet
Right Triangle Trigonometry
Thanksgiving Break
11/26-28
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.
1
Secondary II
Unit 8
2
Secondary II
Unit 8
Unit 8
8-1 Right Triangle Trigonometry
Reminder- Special Right Triangles
30  60  90 Triangle:
45  45  90 Triangle:
60
45
30
Find all missing side lengths.
1.
2.
x
x
6 2
6
45
3.
4.
x
3
30 
y
60 
y
x
6
3
Secondary II
Unit 8
The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of a right triangle.
A
c
b
C
B
a
Pythagorean Theorem demonstrates a relationship that exists between the sides of a right triangle.
a 2  b2  c2
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:
sin  
cosine:
opposite
hypotenuse
cos  
adjacent
hypotenuse
tangent:
tan  
opposite
adjacent
The reciprocals of these functions can also be used to find  . They are defined as followed:
cosecant:
𝑐𝑠𝑐𝜃 =
secant:
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑠𝑒𝑐𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cotangent:
𝑐𝑜𝑡𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
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, C  90
2. r = 5, s = 12, t = 13, T  90
sin A=
sin S=
cos A =
cos S =
tan A =
tan S =
4
Secondary II
Unit 8
Calculator Trigonometry
Round to 3 decimal places!!!
Use your calculator to find the value. Round to 3 decimal places.
3. sin 30 =
4. tan 20 =
5. cos 138 =
6. cos 90 =
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
x = __________
8. cos B = .7878
B = __________
9. tan  = .1648
10. cos R = .450
 = __________
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 = 90
Find
 A and  B and c.
14.
 R = 32 , T  90 , t = 8
Find s, r and S
 A = ______________,
s = _________________,
 B = ______________
r = _________________,
c = _______________
S = ______________
5
Secondary II
16. sin 20 
Unit 8
4
h
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?
6
Secondary II
Unit 8
Additional Notes
7
Secondary II
Unit 8
8-2 Degree Angles, Sketching Angles in Standard Position
y
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.
x
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.
2. 330
1. 140
y
y
x
__________
y
x
__________
4. 270
3. 510
y
x
__________
x
__________
Coterminal Angles:
5.
Give two angles coterminal with 210º
____________ ____________
6.
Give two angles coterminal with 34º
____________ ____________
8
Secondary II
Unit 8
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.
32
2
=
16 = 4
b.
2
75
=
2
553
=
2
5 3

3
3
=
6
(rationalized)
15
Divide the radicals. Rationalize the denominators.
7.
20
5
8.
25
9.
3
3
5x
Special Right Triangles and Trig Ratios
You will frequently need to determine the value of trigonometric ratios for 30 , 60 , and 45 angles on a
coordinate system.
Simplest Side Ratios for Special Right Triangles
45°
2
1
1
√2
30°
√3
45°
1
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
y
y
y
x
x
x
9
Secondary II
Unit 8
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:
For “axis” angles:
y

x




Steps for the following problems:
The “x” value represents the adjacent leg
The “y” value represents the opposite leg
The radius of the circle (length = 1) represents
the hypotenuse of the triangle
Label the points where the circle intersects the x
and the y axis.

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 150
2. cos ( 60 ) = ________
= ________
y
y
x
x
10
Secondary II
Unit 8
4. sin 315 = ________
3. tan 210˚ = ________
y
y
x
x
For the following trig ratios follow the steps above, but use the “Axis” diagram to find the opposite,
adjacent, and hypotenuse values.
5. sin 180 = ________
6. tan 270˚ = ________
y
y
x
x
7. If (3, -2) is a point on a graph, give the sin, cos and tan.
11
Secondary II
Unit 8
12
Secondary II
Unit 8
Additional Notes
13
Secondary II
Unit 8
8-3 Trig Identities
Reciprocal Identities
sin  
csc 
cos 
sec 
tan  
tan  
cot  
cot  
Pythagorean Identities
sin 2   cos 2   1
1  tan 2   sec2 
1  cot 2   csc2 
Prove the following identities:
1. sec 𝑥 cos 𝑥 = 1
2. csc 𝜃 tan 𝜃 = sec 𝜃
2
3. sin  sec  csc   tan 
**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.
14
Secondary II
4. sec   2 tan   (1  tan  )
2
Unit 8
2
5. Simplify.
cos x sin x
(sin x  1)(sin x  1)
15
Secondary II
Unit 8
Additional Notes
16