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