Math 10
Ms. Albarico
Students are expected to:•Demonstrate an understanding of and apply properties to operations involving square roots.
• Relate the trigonometric functions to the ratios in similar right triangles.
• Use calculators to find trigonometric values of angles and angles to find when trigonometric values are known.
* Solve problems using the trigonometric ratios.
A RATIO is a comparison of two numbers. For
example; boys to girls cats : dogs
right : wrong.
In Trigonometry, the comparison is between
sides of a triangle.
We need to do some housekeeping before we can proceed…
In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE.
Two things MUST BE understood:1. This is the hypotenuse.. This will
ALWAYS be the hypotenuse2. This is 90°… this makes the right
triangle a right triangle…. Without it, we can not do this trig… we WILL NOT use it in our calculations because we COULD NOT do calculations without it.
Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other angles and the 2 other sides.A We will refer to the sides
in terms of their proximity to the angle
If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.
opposite
adjacenthypotenuse
B
If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.
opposite
adjacent
hypotenuse
Remember we won’t use the right angle
X
One more thing…
Here we go!!!!
Trigonometric Ratios
Name“say” Sine Cosine tangent
AbbreviationAbbrev. Sin Cos Tan
Ratio of an angle measure
Sinθ = opposite side hypotenuse
cosθ = adjacent side hypotenuse
tanθ =opposite side adjacent side
Values of Trigonometric Function
00 300 450 600 900
Sine 0 0.5 1/2 3/2 1
Cosine 1 3/2 1/2 0.5 0
Tangent 0 1/ 3 1 3 Not defined
Cosecant Not defined 2 2 2/ 3 1
Secant 1 2/ 3 2 2 Not defined
Cotangent Not defined 3 1 1/ 3 0
One more time…Here are the ratios:
sinθ = opposite side hypotenuse
cosθ = adjacent side hypotenuse
tanθ = opposite side adjacent side
B c a
C b A
Write the ratio for sin A
Sin A = a c
Write the ratio for cos A
Cos A = b c
Write the ratio for tan A
Tan A = a b
Let’s switch angles: Find the sin, cos and tan for Angle B:
Sin B = b
cCos B = a
c
Tan B = b
a
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Calculator Commands
Find tan A: 24.19 12
A 21
Tan A = opp/adj = 12/21
Tan A = .5714
8
4A
Tan A = 8/4 = 2 8
Find tan A:
Note:
Given Looking for Use
Ratio of sidesAngle measure
SIN-1
COS-1
TAN-1
Angle, side Missing side SIN, COS, TAN
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Calculator Commands Reminder
To solve for Angles:
A B
C
xo
Opp’
Adj’
adjoppxo tan
Now we need to look at the two ratios involving the hypotenuse:
sin xo = OppositeHypotenuse
hyp’
cos xo = AdjacentHypotenuse
Calculator Commands
For Trigonometric Inverse Functions:
1) Press 2ND, useSIN for SIN-1
COS for COS-1
TAN for TAN-1
Calculate the angle b o below.
14.8cm
9.7cmb o
(1) Identify the two sides marked.a
h
(2) Choose the correct trig ratio .
(3) Substitute in values .
(4) Calculate the ratio(3 decimal places).
(5) Use the inverse cosine function on your calculator to calculate the angle .b o = 49.1o
hypadjxo cos
8.147.9cos ob
655.0cos ob
655.0cos 1ob
Remembering the Trigonometric Ratios:Look again at the three trig ratios given below:
hypoppxo sin
hypadjxo cos
adjoppxo tan
Take the first letter of each word.
Write the letters in order.
S O H C A H T O A
C2cm
B 3cm A
Find an angle that has a tangent (ratio) of 2
3
Round your answer to the nearest degree.
Process:
I want to find an ANGLE.
I was given the sides (ratio).
Tangent is opp
adj
Solution:
TAN-1(2/3) = 34°
Ok… we’ve found side lengths, now let’s find angle measures.
Refer to your table… what function will we use to find angle measures?
SIN-1COS-1TAN-1 These are
called INVERSE FUNCTIONS.
Homework!
In your notebook, CYU # 18, 19, 20, 21, 22, 24, and 25 on pages 239-240.
Class Work
In your notebook, solve the following:
CYU # 12, 13, 14, 15, 16 on pages 236-237.
Work Period
Work with your group members about the final design of your pet house.
Remember :
Your scale drawing must be accurate and precise.
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