Right Triangle
Trigonometry:
Word Splash
Use your prior knowledge or make up a meaning for the following words to create a story. Use your
imagination!
Relating to the
Real World
• Before any spacecraft ever traveled to another
planet, astronomers had figured out the
distance from each planet to the sun. They
accomplished this feat by using trigonometry
the mathematics of triangle measurement. You
will learn how to use trigonometry to measure
distances that you could never otherwise
measure.
Measure for Measure
• What do the Trigonometric Ratios tell you
about the parts of a triangle?
• What are the conditions for triangles to be
similar?
• How can you remember the Trigonometric
ratios?
Label These Two Triangles
Angle A
Angle A
The Tangent Ratio
• The word Trigonometry comes from the Greek
words meaning “triangle measurement.” A ratio of
the lengths of sides of a right triangle. This ratio
is called the TANGENT.
• TANGENT OF
A = leg opposite A
leg adjacent to A
• Slope of a line = Rise
Run
Using the Tangent Ratio
• Tangent of < A = Leg OPPOSITE TO < A
Leg ADAJACENT TO <A
B
THIS EQUATION CAN BE
ABBREVIATED AS:
Leg
Opposite
to < A, BC
TAN A = OPPOSITE = BC
A
C
Leg Adjacent to < A, CA
ADJACENT
CA
Example
Write the Tangent Ratios for < U and < T.
T
Tan U = opposite = TV = 3
adjacent UV 4
5
Tan T = opposite = UV = 4
adjacent TV 3
3
V
4
U
Try This
• Write the Tangent ratios for < K and < J
• How is Tan K related to the Tan J ?
J
3
K
L
7
Try This: Find the Tangent
of < A to the nearest tenth
1.
2.
4
A
8
5
A
Hint: Find the Ratio First!
4
Practice:
• Find the Tan A and Tan B ratios of each triangle.
B
5
5
3
4
2
B
A
A
4
B
7
B
6
A
5
A
10
Using the Sine Ratio
• Sine of < A = Leg OPPOSITE TO < A
Hypotenuse
B
THIS EQUATION CAN BE
ABBREVIATED AS:
Leg
Opposite
to < A, BC
Sin A = OPPOSITE =
BC
HYPOTENUSE AB
A
C
Sin B = OPPOSITE =
Leg Opposite to < B, CA
CA
HYPOTENUSE AB
Example
Write the Sine Ratios for < U and < T.
T
Sin U = opposite = TV = 3
hypotenuse TU 5
5
Sin T = opposite = UV = 4
hypotenuse TU 5
3
V
4
U
Try This
• Write the Sine ratios for < K and < J
J
10
6
K
L
8
Practice:
• Find the Sin A and Sin B ratios of each triangle.
B
4
5
3
4
2
B
A
A
5
B
7
B
8
A
7
A
10
Using the Cosine Ratio
• Cosine of < A = Leg Adjacent TO < A
Hypotenuse
B
THIS EQUATION CAN BE
ABBREVIATED AS:
Leg
Adjacent
to < B, BC
Cos A = ADJACENT = CA
HYPOTENUSE AB
A
C
Cos B = ADJACENT = BC
Leg Adjacent to < A, CA
HYPOTENUSE AB
Example
Write the Cosine Ratios for < U and < T.
T
Cos U = adjacent = UV = 4
hypotenuse TU 5
5
Cos T = adjacent = TV = 3
hypotenuse TU 5
3
V
4
U
Try This
• Write the Cosine ratios for < K and < J
J
10
6
K
L
8
Practice:
• Find the Cos A and Cos B ratios of each triangle.
B
4
5
3
4
2
B
A
A
5
B
7
B
6
A
5
A
10
Finding Angles
• To Find an angle use the inverse function of your
calculator.
B
You may use the
Leg
Opposite
to < A, BC
A
C
Leg Adjacent to < A, CA
Sin-1
Cos-1
or Tan-1
to find any angle.
Practice:
• Use the Cos-1 and Sin -1 ratios to find the angles of
each triangle.
B
5
6
3
4
2
B
A
A
B
7
B
6
A
7
A
10