Transcript 8-3 8-4 Sine Cosine Tangent

Objectives
• Define and apply sine, cosine, and tangent
in right triangles
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Trigonometric Ratios for Right Triangles
By the AA Similarity Postulate, a right triangle with
a given acute angle is similar to every other right
triangle with that same acute angle measure. So
∆ABC ~ ∆DEF ~ ∆XYZ, and
. These are
trigonometric ratios. A trigonometric ratio is a ratio
of two sides of a right triangle.
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Sine, Cosine, Tangent Definitions
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Story of Chief Soh Cah Toa
• A young Indian, 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 and injure himself. 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 toe that he injured by kicking the stone, he
was constantly at the edge of the river, soaking his toe in the cooling
waters. For that behavior, he was named Chief Soh Cah Toa.
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Ex. 1
B
13 hyp
5
opp
A
C
12
adj
Sin A = ?
1. Mark angle A.
2. Label the sides in relation to
angle A (opp, adj, hyp)
3. Circle the sides that we use for
sin
4. Decide which side goes on top
of the fraction
5. Write the fraction Sin A = 5/13
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Ex. 2
B
13 hyp
5
adj
A
C
12
opp
Cos B = ?
1. Mark angle B.
2. Label the sides in relation to
angle B (opp, adj, hyp)
3. Circle the sides that we use for
cos
4. Decide which side goes on top
of the fraction
5. Write the fraction Cos B = 5/13
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Trigonometry with a Calculator Ex. 1
B
13 hyp
x
opp
34°
C
adj
A
x=?
1. Mark angle A.
2. Label the sides in relation to
angle A (opp, adj, hyp)
3. Decide which trig. func. You
can use.
4. Decide which side goes on top
of the fraction
5. Write the fraction
Sin 34 = x/13
6. Solve for x.
x = 13 * sin 34
x = 7.3
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Trigonometry with a Calculator Ex. 4
B
x
20
opp
52°
C
adj
hyp
A
x=?
1. Mark angle A.
2. Label the sides in relation to
angle A (opp, adj, hyp)
3. Decide which trig. func. You
can use.
4. Decide which side goes on top
of the fraction
5. Write the fraction
Sin 52 = 20/x
6. Solve for x.
x = 20/sin 52
x = 25.4
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Inverse Trigonometric Functions
If you know the sine, cosine, or tangent of an
acute angle measure, you can use the
inverse trigonometric functions to find the
measure of the angle.
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Finding an angle using Trig. Ex. 1
B
14 hyp
5
opp
x
C
adj
A
x=?
1. Mark angle A.
2. Label the sides in relation to
angle A (opp, adj, hyp)
3. Decide which trig. func. You
can use.
4. Decide which side goes on top
of the fraction
5. Write the fraction
Sin x = 5/14
6. Solve for x.
x = sin-1(5/14)
x = 21°
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Finding an angle using Trig. Ex. 2
B
hyp
5
opp
C
x
14
adj
A
x=?
1. Mark angle A.
2. Label the sides in relation to
angle A (opp, adj, hyp)
3. Decide which trig. func. You
can use.
4. Decide which side goes on top
of the fraction
5. Write the fraction
Tan x = 5/14
6. Solve for x.
x = tan-1(5/14)
x = 20°
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