Trigonometric Ratios

1
Trigonometric Ratios

• Please view this tutorial and answer the
  follow-up questions on loose leaf to turn in to
  your teacher.

2
Identifying Parts of a Right Triangle
A

• Hypotenuse always across from the 90 angle
• Side Opposite always across from the angle
  being referenced
• Side Adjacent- always touching the angle being
  referenced
• Note that all angles are marked with capitol
  letters and sides are marked with lower case
  letters

B
C
Angle C measures 90
3
Identifying Parts of a Right Triangle

• What side is opposite of angle A?
• Side BC
• What side is opposite of angle B?
• Side AC
• What side is adjacent to angle A?
• Side AC
• What side is adjacent to angle B?
• Side BC
• What side is the hypotenuse?
• Side AB

A
B
C
4
Trigonometric Ratios (only apply to right
triangles)

• Sine (abbreviated sin)
• Sin x

• Example

A
C
B
Sin A
5
Trigonometric Ratios (only apply to right
triangles)

• Cosine (abbreviated cos)
• Cos x

• Example

A
C
B
Cos A
6
Trigonometric Ratios (only apply to right
triangles)

• Tangent (abbreviated tan)
• Tan x

• Example

A
C
B
Tan A
7
Helpful Hint to Remember the Trig Ratios

• SOH (sine opposite / hypotenuse)
• CAH (cosine adjacent / hypotenuse)
• TOA (tangent opposite / adjacent)
• Remember SOH CAH TOA

8
Time to Practice

• Identify the following trig ratio values

C
3
B
Sin A Sin B Cos A Cos B Tan A
Tan B
4
5
A
9
Time to Practice

• Identify the following trig ratio values

C
3
Sin A Sin B Cos A Cos B Tan
A Tan B
B
4
5
A
10
More Practice

• Identify the following trig ratio values

B
Sin A Sin B Cos A Cos B Tan A
Tan B
13
5
C
A
12
11
More Practice

• Identify the following trig ratio values

B
Sin A Sin B Cos A Cos B Tan
A Tan B
13
5
C
A
12
12
How to use the trig ratios to find missing sides

• Step 1 Make sure your calculator is in degree
  mode
• Step 2 Label the right triangle with the words
  opposite, adjacent, and hypotenuse based on the
  given angle (Note Do not use the right angle.)
• Step 3 From the given information, determine
  which trig ratio should be used to find the side
  length
• Step 4 Substitute in the given information

13
How to use the trig ratios to find missing sides
(continued)

• Step 5 Put a 1 under the trig ratio
• Step 6 Cross multiply
• Step 7 When x, put problem into your calculator
  (Note you may have to divide first to get x by
  itself)
• (NOTE The angles of a triangle MUST add up to be
  180)

14
Example

• Given the following triangle, solve for x.

60
8 cm
x
15
Lets Talk Through the Steps

• Step 1 Check calculator for degree mode
• Press the Mode button and make sure Degree is
  highlighted as in the picture below

16
Step 2

• Label the triangle according to the given angle

60
8 cm- HYPOTENUSE
X - OPPOSITE
17
Step 3

• Identify the trig ratio we should use to solve
  for x.

60
8 cm- HYPOTENUSE
From the 60 angle, we know the hypotenuse and
need to find the opposite. So we need to use
SINE.
X - OPPOSITE
18
Step 4

• Substitute in the given information into the
  equation.

60
8 cm- HYPOTENUSE
Sin x Sin 60
X - OPPOSITE
19
Step 5

• Put a 1 under the trig ratio

60
8 cm- HYPOTENUSE
Sin x Sin 60 1
X - OPPOSITE
20
Step 6

• Cross multiply to solve for x

Sin 60 1
8 sin (60) x
21
Step 7

• Since x is already by itself, I can enter the
  information into the calculator.

Therefore, we can state that x6.93.
22
Lets Look at Another Example

• Suppose that when we set-up the ratio equation,
  we have the following
• Tan 20

23
What Happens When We Cross Multiply?

• Tan 20
• 1
• X tan 20 4 (How do we get x by
  itself?)
• tan 20 tan 20 (Now we have to
  divide by tan 20 in order to solve for x)
• X 4
• tan 20
• X 10.99

24
How to use the trig ratios to find missing angles

• Step 1 Make sure your calculator is in degree
  mode (See slide 15)
• Step 2 Label the right triangle with the words
  opposite, adjacent, and hypotenuse based on the
  given angle (Note Do not use the right angle.)
• Step 3 From the given information, determine
  which trig ratio should be used to find the side
  length
• Step 4 Substitute in the given information

25
How to use the trig ratios to find missing sides
(continued)

• Step 5 Solve for x by taking the inverse
  (opposite operation) of the trig ratio.
• Step 6 When x, put problem into your
  calculator.

26
Calculator Steps for Finding Angles

• To solve for x, remember to take the inverse trig
  function.
• On the calculator, you can find the inverse trig
  functions by pressing 2nd and then the trig
  function.

27
Lets Look at an Example

• Given the following triangle, solve for x.

62 cm
90
x
200 cm
28
Step 2

• Label the sides opposite, adjacent, or hypotenuse
  from angle x.

OPPOSITE
62 cm
90
200 cm
HYPOTENUSE
x
29
Step 3

• Since we have the opposite and the hypotenuse, we
  need to use SINE.

OPPOSITE
62 cm
90
200 cm
HYPOTENUSE
x
30
Step 4

• Substitute in the given information into the
  equation.

OPPOSITE
62 cm
Sin x
90
200 cm
HYPOTENUSE
x
31
Step 5

• To solve for x, we need to take the inverse of
  sine on both sides.

OPPOSITE
62 cm
Sin x
90
Sin-1 (sin x) Sin-1
200 cm
HYPOTENUSE
x
32
Step 6

• Now just type in the x on your calculator.

Sin x
Sin-1 (sin x) Sin-1
X Sin-1
X 18
33
Now Its Your Turn!

• Use what youve just reviewed to help you answer
  the following questions.
• Submit all of your work to your teacher after
  completing the tutorial.
• Dont be afraid to go back through the slides if
  you get stuck.
• GOOD LUCK!

34
Problem 1

• Complete the following ratios.

Sin A Sin B Cos A Cos B Tan
A Tan B
6 cm
C
A
90
8 cm
10 cm
B
35
Problem 2

• Solve for x and y.

40 ft
90
y
55
x
36
Problem 3

• Solve for angles A and B.

A
5 in
B
90
7 in
C