Trigonometric Ratios

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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 ( right triangle).
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CCSS G.SRT.7

• EXPLAIN and USE the relationship between the
  sine and cosine of complementary angles.

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Standards for Mathematical Practice

• 1. Make sense of problems and persevere in
  solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
  reasoning of others.  
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning.

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Warm up

• Solve the equations
• A) 0.875 x/18
• B) 24/y .5
• C) y/25 .96

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E.Q

• How can we find the sin, cosine, and the tangent
  of an acute angle?
• How do we use trigonometric ratios to solve
  real-life problems?

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Trig. Ratios
Name say Sine Cosine tangent
Abbreviation Abbrev. Sin Cos Tan
Ratio of an angle measure Sin? opposite side hypotenuse cos? adjacent side hypotenuse tan? opposite side adjacent side
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Three Trigonometric Ratios

• Sine abbreviated sin.
• Ratio sin ? opposite side
• hypotenuse
• Cosine - abbreviated cos.
• Ratio cos ? adjacent side
• hypotenuse
• Tangent - abbreviated tan.
• Ratio tan ? opposite side
• adjacent side

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Lets practice
Write the ratio for sin A Sin A o a
h c Write the ratio for cos A Cos A
a b h c Write the ratio
for tan A Tan A o a a b
B c a C b
A
Lets switch angles Find the sin, cos and tan
for Angle B
Tan B b a

Sin B b c
Cos B a c

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Make sure you have a calculator
I want to find Use these calculator keys
sin, cos or tan ratio SIN COS TAN
Angle measure SIN-1 COS-1 TAN-1
Set your calculator to Degree.. MODE (next to
2nd button) Degree (third line down highlight
it) 2nd Quit
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Lets practice
Find an angle that has a tangent (ratio) of 2
3 Round your answer to the
nearest degree.
C 2cm B 3cm A
Process I want to find an ANGLE I was given the
sides (ratio) Tangent is opp
adj TAN-1(2/3) 34
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Practice some more
Find tan A 24.19
12 A 21
Tan A opp/adj 12/21 Tan A .5714
Find tan A
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Tan A 8/4 2
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Trigonometric Ratios

• When do we use them?
• On right triangles that are NOT 45-45-90 or
  30-60-90

Find tan 45 1 Why? tan opp hyp
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Using trig ratios in equations

• Remember back in 1st grade when you had to solve
• 12 x What did you do?
• 6

(6) (6)
72 x
Remember back in 3rd grade when x was in the
denominator? 12 6 What
did you do? x
(x) (x)
12x 6
__ __ 12 12
x 1/2
14

• Ask yourself
• In relation to the angle, what pieces do I have?

Opposite and hypotenuse
Ask yourself What trig ratio uses Opposite and
Hypotenuse?
SINE
Set up the equation and solve
Sin 34 x 15
(15) (15)
(15)Sin 34 x
8.39 cm x
15

• Ask yourself
• In relation to the angle, what pieces do I have?

53
12 cm
Opposite and adjacent
Ask yourself What trig ratio uses Opposite and
adjacent?
x cm
tangent
Set up the equation and solve
Tan 53 x 12
(12) (12)
(12)tan 53 x
15.92 cm x
16

• Ask yourself
• In relation to the angle, what pieces do I have?

x cm
Adjacent and hypotenuse
68
Ask yourself What trig ratio uses adjacent and
hypotnuse?
18 cm
cosine
Set up the equation and solve
(x) (x)
Cos 68 18 x
(x)Cos 68 18
_____ _____ cos 68 cos 68
X 18 cos 68
X 48.05 cm
17

• This time, youre looking for theta.
• Ask yourself
• In relation to the angle, what pieces
• do I have?

42 cm
22 cm
Opposite and hypotenuse
THIS IS IMPORTANT!!
Ask yourself What trig ratio uses opposite and
hypotenuse?
?
sine
Set up the equation (remember youre looking for
theta)
Sin ? 22 42
Remember to use the inverse function when you
find theta
Sin -1 22 ? 42
31.59 ?
18

• Youre still looking for theta.

?
THIS IS IMPORTANT!!
Ask yourself What trig ratio uses the parts I
was given?
22 cm
17 cm
tangent
Set it up, solve it, tell me what you get.
tan ? 17 22
tan -1 17 ? 22
37.69 ?
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Using trig ratios in equations

• Remember back in 1st grade when you had to solve
• 12 x What did you do?
• 6

(6) (6)
72 x
Remember back in 3rd grade when x was in the
denominator? 12 6 What
did you do? x
(x) (x)
12x 6
__ __ 12 12
x 1/2
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Types of Angles

• The angle that your line of sight makes with a
  line drawn horizontally.
• Angle of Elevation
• Angle of Depression

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Indirect Measurement
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SOA CAH TOA

• SOA CAH TOA

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Solving a right triangle

• Every right triangle has one right angle, two
  acute angles, one hypotenuse and two legs. To
  solve a right triangle, means to determine the
  measures of all six (6) parts. You can solve a
  right triangle if the following one of the two
  situations exist
• Two side lengths
• One side length and one acute angle measure

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E.Q

• How do we use right triangles to solve real life
  problems?

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Note

• As you learned in Lesson 9.5, you can use the
  side lengths of a right triangle to find
  trigonometric ratios for the acute angles of the
  triangle. As you will see in this lesson, once
  you know the sine, cosine, or tangent of an acute
  angle, you can use a calculator to find the
  measure of the angle.

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WRITE THIS DOWN!!!

• In general, for an acute angle A
• If sin A x, then sin-1 x m?A
• If cos A y, then cos-1 y m?A
• If tan A z, then tan-1 z m?A

The expression sin-1 x is read as the inverse
sine of x.

• On your calculator, this means you will be
  punching the 2nd function button usually in
  yellow prior to doing the calculation. This is
  to find the degree of the angle.

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Example 1

• Solve the right triangle. Round the decimals to
  the nearest tenth.

HINT Start by using the Pythagorean Theorem.
You have side a and side b. You dont have the
hypotenuse which is side cdirectly across from
the right angle.
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Example 1
(hypotenuse)2 (leg)2 (leg)2
Pythagorean Theorem
Substitute values
c2 32 22
Simplify
c2 9 4
Simplify
c2 13
Find the positive square root
c v13
Use a calculator to approximate
c 3.6
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Example 1 continued

• Then use a calculator to find the measure of ?B
• 2nd function
• Tangent button
• 2
• Divided by symbol
• 3 33.7

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Finally

• Because ?A and ?B are complements, you can write
• m?A 90 - m?B 90 - 33.7 56.3
• ?The side lengths of the triangle are 2, 3 and
  v13, or about 3.6. The triangle has one right
  angle and two acute angles whose measure are
  about 33.7 and 56.3.

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Ex. 2 Solving a Right Triangle (h)

• Solve the right triangle. Round decimals to the
  nearest tenth.

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You are looking for opposite and hypotenuse which
is the sin ratio.
Set up the correct ratio
Substitute values/multiply by reciprocal
Substitute value from table or calculator
13(0.4226) h
Use your calculator to approximate.
5.5 h
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Ex. 2 Solving a Right Triangle (g)

• Solve the right triangle. Round decimals to the
  nearest tenth.

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You are looking for adjacent and hypotenuse which
is the cosine ratio.
Set up the correct ratio
Substitute values/multiply by reciprocal
Substitute value from table or calculator
13(0.9063) g
11.8 h
Use your calculator to approximate.
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Using Right Triangles in Real Life

• Space Shuttle During its approach to Earth, the
  space shuttles glide angle changes.
• A. When the shuttles altitude is about 15.7
  miles, its horizontal distance to the runway is
  about 59 miles. What is its glide angle? Round
  your answer to the nearest tenth.

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Solution
Glide ? x
15.7 miles

• You know opposite and adjacent sides. If you
  take the opposite and divide it by the adjacent
  sides, then take the inverse tangent of the
  ratio, this will yield you the slide angle.

59 miles
opp.
tan x
Use correct ratio
adj.
15.7
Substitute values
tan x
59
Key in calculator 2nd function, tan 15.7/59
14.9
? When the space shuttles altitude is about
15.7 miles, the glide angle is about 14.9.
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B. Solution
Glide ? 19
h

• When the space shuttle is 5 miles from the
  runway, its glide angle is about 19. Find the
  shuttles altitude at this point in its descent.
  Round your answer to the nearest tenth.

5 miles
opp.
tan 19
Use correct ratio
adj.
h
Substitute values
tan 19
5
h
5
Isolate h by multiplying by 5.
5 tan 19
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? The shuttles altitude is about 1.7 miles.
1.7 h
Approximate using calculator