Ratios in Right Triangles

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Ratios in Right Triangles
WHAT YOU WILL LEARN

• To find trigonometric ratios using right
  triangles, and
• To solve problems using trigonometric ratios.

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DEFINITIONS

• Trigonometry The study of the properties of
  triangles and trigonometric functions and their
  applications.
• Trigonometric ratio A ratio of the measures of
  two sides of a right triangle is called a
  trigonometric ratio.

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Great Chief Soh Cah Toa
A young brave, frustrated by his inability to
understand the geometric constructions of his
tribe's battle dress, kicked out in anger against
a stone and crushed his big toe. Fortunately, he
learned from this experience, and began to use
study and concentration to solve his problems
rather than violence. This was especially
effective in his study of math, and he went on to
become the wisest man of his tribe. He studied
many aspects of trigonometry and even today we
remember many of the functions by his name. When
he became an adult, the tribal priest gave him a
name that reflected his special nature -- one
that reminded them of his great discoveries and
of the event which changed his life. Because he
was troubled throughout his life by the
problematic foot, he was constantly at the edge
of the river, soaking his aches in the cooling
waters. For that behavior, he was named Chief
Soh Cah Toa.
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DEFINITIONS

• Sine Opposite side over hypotenuse.
• Cosine Adjacent side over hypotenuse.
• Tangent Opposite side over hypotenuse.

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Example
Find the sin S, cos S, tan S, sin E, cos E and
tan E. Express each ratio as a fraction and as a
decimal.
M
Sin S ME/SE 3/5 or 0.6
Cos S SM/SE 4/5 or 0.8
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3
Tan S ME/SM ¾ or 0.75
Sin E SM/SE 4/5 or 0.8
E
S
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Cos E ME/SE 3/5 or 0.6
Tan E SM/ME 4/3 or 1.3
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Example
Find each value using a calculator. Round to the
nearest ten thousandths.

• Cos 41
• Sin 78

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Example
A plane is one mile about sea level when it
begins to climb at a constant angle of 2 for the
next 70 ground miles. How far above sea level is
the plane after its climb?
2
h
1 mi Sea level
70 mi
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Example
Find m A in right triangle ABC for A(1,2),
B(6,2), and C(5,4).
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The End!
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