Download RT-4 Trig Ratios POWERPOINT

8-3 8-4 Sine, Cosine
and Tangent Ratios
Objective
Students will be able to use sine, cosine, and tangent ratios to determine side
lengths in triangles.
Trigonometric Ratios
• We use the Pythagorean Theorem when we are given two
sides of a right triangle and we want to find the third.
• We will use trigonometric ratios when we are given one side
and one angle (other than the 90 degree angle) of a right
triangle and want to find another (or both) side(s).
• There are three trigonometric ratios we will utilize
• Sine (abbreviated sin)
• Cosine (abbreviated cos)
• Tangent (abbreviated tan)
Side Definitions
• Opposite side: leg directly across from the angle of
interest
• Adjacent side: leg next to the angle of interest
• Hypotenuse: side directly across from the right angle
• The opposite and adjacent sides differ depending on the
angle of interest. For example, if you are looking at angle
X, then the opposite side is a. However, if you are
looking at angle Y, then the opposite side is b.
Other notes
• Never use the right angle when using trigonometric ratios.
Only use one of the two acute angles.
• Calculators must be in “degree” mode. To check, press the
“mode” button and go down to where you see “radian” and
“degree”. If not already highlighted, highlight “degree” and
press enter.
• If your calculator is in “radian” mode, you will not get the
correct answers we are looking for here. (They are correct
answers but for Geometry we want answers in Degree Mode)
Example 1:
Write the sine, cosine, and tangent ratios for angles T and U.
You Try
Write the sine, cosine, and tangent ratios for angles J and K.
Finding Side Length
• When using
trigonometry to solve
for a side length, first
determine which trig
ratio to use based on
the given information.
• Then, substitute in the
information.
• Finally, solve as you
would solve a
proportion. We
usually round to the
nearest tenth.
Example 2: Find the value of x
to the nearest tenth.
1)
2)
opp
sin 40 =
hyp
x
sin 40 =
12
12sin 40 = x
opp
tan 56 =
adj
x
tan 56 =
12
12 tan 56 = x
7.7 » x
17.8 » x
Example 2b: Find the value of
x to the nearest tenth.
3)
adj
hyp
x
cos 24 =
14
14 cos24 = x
cos 24 =
12.8 » x
4)
opp
hyp
x
sin 64 =
7
7sin 64 = x
sin 64 =
6.3 » x
Try these
5)
adj
cos35 =
hyp
x
cos35 =
19
19 cos35 = x
6)
15.6 » x
7)
opp
sin19 =
hyp
x
sin19 =
10
10sin19 = x
3.3 » x
opp
tan 54 =
hyp
x
tan 54 =
10
10 tan 54 = x
13.8 » x
What if my variable is in the
Denominator?
8)
9)
opp
hyp
9
sin 23 =
x
xsin 23 = 9
sin 23 =
9
x=
sin 23
x » 23.0
opp
tan 75 =
adj
21
tan 75 =
x
xtan 75 = 21
21
tan 75
5.63 » x
x=
Fun Stuff!
10)
adj
hyp
9
cos37 =
x
xcos37 = 9
11)
cos37 =
9
x=
cos37
x » 11.27
opp
hyp
8
sin15 =
x
xsin15 = 8
sin15 =
8
x=
sin15
x » 30.90
Just some more…
12)
13)
adj
hyp
32
cos 61 =
x
xcos61 = 32
opp
tan 41 =
adj
23
tan 41 =
x
xtan 41 = 23
cos 61 =
23
x=
tan 41
x » 26.46
32
x=
cos61
x » 66.01
Finding Angle Measures
• If given two sides of a right triangle, we can
determine the angle measures by using inverse
trigonometric ratios.
• Start by determining the appropriate ratio to use
and substituting in your information.
• Then, take the inverse of the ratio.
• To do this on the calculator, hit “2nd” and then
hit either sin, cos, or tan (depending on which
ratio is appropriate given the problem).
Ex: Finding Angle Measure
Examples: Finding Angles
Find the value of x. Round answers to the nearest degree.
1)
2)
adj
cos x =
hyp
10
cos x =
18
æ 10 ö
x = cos-1 ç ÷
è 18 ø
tan x =
x » 56
x » 29
opp
adj
11
tan x =
20
æ 11 ö
x = tan -1 ç ÷
è 20 ø
Examples: Finding Angles
3)
opp
hyp
12
sin x =
22
æ 12 ö
x = sin -1 ç ÷
è 22 ø
4)
sin x =
cos x =
x » 33
x » 68
adj
hyp
10
cos x =
27
æ 10 ö
x = cos-1 ç ÷
è 27 ø
Try these…
5)
6)
opp
tan x =
adj
100
tan x =
41
æ 100 ö
x = tan -1 ç
÷
è 41 ø
opp
sin x =
hyp
6.5
sin x =
10
-1 æ 6.5 ö
x = sin ç ÷
è 10 ø
x » 68
x » 41
Critical Thinking
• Describe and create a triangle where:
sinÐ = cosÐ
Critical Thinking
• Given the right triangle ABC where C is the Right Angle,
determine if the following statement is valid. Explain why or
why not.
sin A > tan A