right_triangle_trig.ppt

Right Triangle
Trigonometry
Digital Lesson

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

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

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

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

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

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

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

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

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 

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

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

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

right_triangle_trig.ppt

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