Transcript Trigonometric Ratios - Ridley Coreplus Tutorials!

Trigonometric
Ratios
Please view this tutorial and answer the
follow-up questions on loose leaf to turn in
to your teacher.
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
C
B
Angle C measures 90°
Identifying Parts of a Right
Triangle
 What side is opposite of angle A?
 Side BC
A
 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
C
B
Trigonometric Ratios (only apply
to right triangles)
 Sine (abbreviated sin)
 Sin x° =
opposite
hypotenuse
 Example:
A
C
B
Sin A =
BC
AB
Trigonometric Ratios (only apply
to right triangles)
 Cosine (abbreviated cos)
 Cos x° =
adjacent
hypotenuse
 Example:
A
C
B
Cos A =
AC
AB
Trigonometric Ratios (only apply
to right triangles)
 Tangent (abbreviated tan)
 Tan x° =
opposite
adjacent
 Example:
A
C
B
Tan A =
BC
AC
Helpful Hint to Remember the
Trig Ratios
 SOH (sine = opposite / hypotenuse)
 CAH (cosine = adjacent / hypotenuse)
 TOA (tangent = opposite / adjacent)
 Remember SOH CAH TOA
Time to Practice
 Identify the following trig ratio values
C
4
3
B
5
Sin A =
Cos A =
Sin B=
Cos B=
A
Tan A =
Tan B=
Time to Practice
 Identify the following trig ratio values
C
4
3
B
Sin A =
3
5
Sin B=
4
5
5
Cos A =
4
5
Tan A =
3
4
A
Cos B=
Tan B=
3
5
4
3
More Practice
 Identify the following trig ratio values
B
13
A
5
Sin A =
Cos A =
Sin B=
Cos B=
C
12
Tan A =
Tan B=
More Practice
 Identify the following trig ratio values
B
13
A
Sin A =
5
13
Cos A =
12
13
Tan A =
5
12
5
C
Sin B=
Cos B=
12
Tan B=
12
13
5
13
12
5
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
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°)
Example
 Given the following triangle, solve for x.
60°
8 cm
x
Let’s 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
Step 2
 Label the triangle according to the given angle
60°
8 cm- HYPOTENUSE
X - OPPOSITE
Step 3
 Identify the trig ratio we should use to solve for x.
60°
8 cm- HYPOTENUSE
X - OPPOSITE
From the 60° angle,
we know the
hypotenuse and need
to find the opposite.
So we need to use
SINE.
Step 4
 Substitute in the given information into the equation.
60°
8 cm- HYPOTENUSE
Sin x°=
X - OPPOSITE
Sin 60° =
opposite
hypotenuse
x
8
Step 5
 Put a 1 under the trig ratio
60°
8 cm- HYPOTENUSE
Sin x°=
opposite
hypotenuse
X - OPPOSITE
Sin 60° =
1
x
8
Step 6
 Cross multiply to solve for x
Sin 60° =
1
x
8
8 sin (60°) = x
Step 7
 Since x is already by itself, I can enter the information into
the calculator.
Therefore, we can
state that x=6.93.
Let’s Look at Another Example
 Suppose that when we set-up the ratio equation, we have
the following:

Tan 20° =
4
x
What Happens When We Cross
Multiply?

Tan 20° =
4
x
1
X tan 20° =
tan 20°
X=
4
(How do we get x by itself?)
tan 20°
(Now we have to divide by tan 20° in order to solve for x)
4
tan 20°
X = 10.99
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
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.
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.
sin
1
cos
tan
1
1
Let’s Look at an Example
 Given the following triangle, solve for x.
62 cm
90°
x
200 cm
Step 2
Label the sides opposite, adjacent, or hypotenuse from angle
x.
OPPOSITE
62 cm
90°
200 cm
HYPOTENUSE
x
Step 3
Since we have the opposite and the hypotenuse, we need to
use SINE.
OPPOSITE
62 cm
90°
200 cm
HYPOTENUSE
x
Step 4
Substitute in the given information into the equation.
OPPOSITE
62 cm
Sin x =
90°
200 cm
HYPOTENUSE
x
62
200
Step 5
To solve for x, we need to take the inverse of sine on both
sides.
OPPOSITE
62 cm
Sin x =
90°
200 cm
HYPOTENUSE
x
62
200
Sin-1 (sin x) = Sin-1
 62 


200 
Step 6
Now just type in the x= on your calculator.
Sin x =
62
200
Sin-1 (sin x) = Sin-1
X = Sin-1
X = 18°
 62 


200 
 62 


200 
Now It’s Your Turn!
 Use what you’ve just reviewed to help you answer the
following questions.
 Submit all of your work to your teacher after completing
the tutorial.
 Don’t be afraid to go back through the slides if you get
stuck.
 GOOD LUCK!
Problem #1
 Complete the following ratios.
6 cm
C
90°
8 cm
10 cm
B
Sin A =
Sin B =
A
Cos A=
Cos B=
Tan A=
Tan B=
Problem #2
 Solve for x and y.
y
90°
55°
x
40 ft
Problem #3
 Solve for angles A and B.
A
5 in
B
90°
7 in
C