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Similar to right_triangle_trig.ppt
Similar to right_triangle_trig.ppt (20)
right_triangle_trig.ppt
- 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
The six trigonometric functions of a right triangle, with an acute
angle , are defined by ratios of two sides of the triangle.
The sides of the right triangle are:
the side opposite the acute angle ,
the side adjacent to the acute angle ,
and the hypotenuse of the right triangle.
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
adj
hyp
θ
sin = cos = tan =
csc = sec = cot =
opp
hyp
adj
hyp
hyp
adj
adj
opp
opp
adj
hyp
opp
- 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
Calculate the trigonometric functions for .
The six trig ratios are
4
3
5
sin =
5
4
tan =
3
4
sec =
3
5
cos =
5
3
cot =
4
3
csc =
4
5
- 4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
Geometry of the 45-45-90 triangle
Consider an isosceles right triangle with two sides of
length 1.
1
1
45
45
2
2
1
1 2
2
The Pythagorean Theorem implies that the hypotenuse
is of length .
2
- 5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
Calculate the trigonometric functions for a 45 angle.
2
1
1
45
csc 45 = = =
1
2 2
opp
hyp
sec 45 = = =
1
2 2
adj
hyp
cos 45 = = =
2
2
2
1
hyp
adj
sin 45 = = =
2
2
2
1
hyp
opp
cot 45 = = = 1
opp
adj
1
1
tan 45 = = = 1
adj
opp
1
1
- 6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
60○ 60○
Consider an equilateral triangle with
each side of length 2.
The perpendicular bisector
of the base bisects the
opposite angle.
The three sides are equal, so the
angles are equal; each is 60.
Geometry of the 30-60-90 triangle
2 2
2
1 1
30○ 30○
3
Use the Pythagorean Theorem to
find the length of the altitude, .
3
- 7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
Calculate the trigonometric functions for a 30 angle.
1
2
30
3
csc 30 = = = 2
1
2
opp
hyp
sec 30 = = =
3
2
3
3
2
adj
hyp
cos 30 = =
2
3
hyp
adj
tan 30 = = =
3
1
3
3
adj
opp
cot 30 = = =
1
3 3
opp
adj
sin 30 = =
2
1
hyp
opp
- 8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
Calculate the trigonometric functions for a 60 angle.
1
2
60○
3
csc 60 = = =
3
2
3
3
2
opp
hyp
sec 60 = = = 2
1
2
adj
hyp
cos 60 = =
2
1
hyp
adj
tan 60 = = =
1
3
3
adj
opp
cot 60 = = =
3
1
3
3
opp
adj
sin 60 = =
2
3
hyp
opp
- 9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
Trigonometric Identities are trigonometric
equations that hold for all values of the variables.
Example: sin = cos(90 ), for 0 < < 90
Note that and 90 are complementary
angles.
Side a is opposite θ and also
adjacent to 90○– θ .
a
hyp
b
θ
90○– θ
sin = and cos (90 ) = .
So, sin = cos (90 ).
b
a
b
a
- 10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
Fundamental Trigonometric Identities for 0 < < 90.
Cofunction Identities
sin = cos(90 ) cos = sin(90 )
tan = cot(90 ) cot = tan(90 )
sec = csc(90 ) csc = sec(90 )
Reciprocal Identities
sin = 1/csc cos = 1/sec tan = 1/cot
cot = 1/tan sec = 1/cos csc = 1/sin
Quotient Identities
tan = sin /cos cot = cos /sin
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2
- 11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
Example:
Given sin = 0.25, find cos , tan , and sec .
cos = = 0.9682
tan = = 0.258
sec = = 1.033
1
9682
.
0
9682
.
0
25
.
0
0.9682
1
Use the Pythagorean Theorem to solve for
the third side.
0.9682
Draw a right triangle with acute angle , hypotenuse of length
one, and opposite side of length 0.25.
θ
1
0.25
- 12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
Example: Given sec = 4, find the values of the
other five trigonometric functions of .
Use the Pythagorean Theorem to solve
for the third side of the triangle.
tan = = cot =
1
15
15
1
15
sin = csc = =
4
15
15
4
sin
1
cos = sec = = 4
4
1
cos
1
15
θ
4
1
Draw a right triangle with an angle such
that 4 = sec = = .
adj
hyp
1
4
- 13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
Example:
Given sin = 0.25, find cos , tan , and sec .
cos = = 0.9682
tan = = 0.258
sec = = 1.033
1
9682
.
0
9682
.
0
25
.
0
0.9682
1
Use the Pythagorean Theorem to solve for
the third side.
0.9682
Draw a right triangle with acute angle , hypotenuse of length
one, and opposite side of length 0.25.
θ
1
0.25