Right Triangle Trigonometry
Trigonometry is based upon ratios of the sides of
right triangles.
The six trigonometric functions of a right triangle,
with an acute angle , are defined by ratios of two sides
of the triangle.
hyp
opp
θ
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.
adj
Right Triangle Trigonometry
The hypotenuse is the longest side and is always
opposite the right angle.
The opposite and adjacent sides refer to another angle,
other than the 90o.
A
A
Trigonometric Ratios
hyp
opp
θ
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
cot
S OH C AH T OA
= adj
opp
Reciprocal Functions
sin  = 1/csc 
cos  = 1/sec 
tan  = 1/cot 
csc  = 1/sin 
sec  = 1/cos 
cot  = 1/tan 
Finding the ratios
The simplest form of question is finding the decimal
value of the ratio of a given angle.
Find using calculator:
1) sin 30
=
sin
30
2) cos 23
=
3) tan 78
=
4) tan 27
=
5) sin 68
=
=
Using ratios to find angles
It can also be used in reverse, finding an angle from a ratio.
To do this we use the sin-1, cos-1 and tan-1 function keys.
Example:
1. sin x = 0.1115 find angle x.
sin-1
(
shift
0.1115
sin
=
)
x = sin-1 (0.1115)
x = 6.4o
2. cos x = 0.8988 find angle x
cos-1
(
shift
0.8988
cos
)
x = cos-1 (0.8988)
x = 26o
=
Calculate the trigonometric functions for  .
Calculate the trigonometric functions for .
5
The six trig ratios are
sin
cos
tan
cot
sec
csc
4
=
5
3
=
5
4
=
3
3
=
4
5
=
3
5
=
4
3
sin α =
5
4
cos α =
5
3
tan α =
4
4
cot α =
3
5
sec α =
4
5
csc α =
3

4

3
What is the
relationship of
α and θ?
They are
complementary
(α = 90 – θ)
Cofunctions
sin  = cos (90  )
sin  = cos (π/2  )
cos  = sin (90  )
cos  = sin (π/2  )
tan  = cot (90  )
tan  = cot (π/2  )
cot  = tan (90  )
cot  = tan (π/2  )
sec  = csc (90  ) csc  = sec (90  )
sec  = csc (π/2  ) csc  = sec (π/2  )
Finding an angle from a triangle
To find a missing angle from a right-angled triangle we
need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan
and use the calculator to identify the angle from the
decimal value of the ratio.
1.
Find angle C
14 cm
C
6 cm
a) Identify/label the names of
the sides.
b) Choose the ratio that
contains BOTH of the
letters.
1.
We have been given the
adjacent and hypotenuse so
we use COSINE:
h
14 cm
adjacent
Cos A = hypotenuse
C
6 cm
a
a
h
Cos C = 6
14
Cos A =
Cos C = 0.4286
C = cos-1 (0.4286)
C = 64.6o
2. Find angle x
Given adj and opp
need to use tan:
x
3 cma
Tan A =
o 8 cm
Tan A =
o
a
Tan x =
8
3
Tan x = 2.6667
x = tan-1 (2.6667)
x = 69.4o
opposite
adjacent
3.
10 cm
12 cm
Given opp and hyp
need to use sin:
opposite
Sin A = hypotenuse
y
o
h
sin x = 10
12
sin A =
sin x = 0.8333
x = sin-1 (0.8333)
x = 56.4o
Finding a side from a triangle
4.
7 cm
k
30o
We have been given
the adj and hyp so we
use COSINE:
adjacent
Cos A =
hypotenuse
Cos A = a
h
Cos 30 = k
7
Cos 30 x 7 = k
6.1 cm = k
5.
We have been given the opp
and adj so we use TAN:
50o
4 cm
Tan A =
r
Tan A =
Tan 50 =
Tan 50 x 4 = r
4.8 cm = r
o
a
r
4
6.
k
We have been given the opp
and hyp so we use SINE:
12 cm
Sin A =
25o
sin A =
sin 25 =
Sin 25 x 12 = k
5.1 cm = k
o
h
k
12
1.
x
30o
Cos A =
a
h
Cos 30 =
5
x
5
cos 30
x =
5 cm
x = 5.8 cm
2.
Tan A =
50o
Tan 50 =
4 cm
o
a
r
4
Tan 50 x 4 = r
4.8 cm = r
r
3.
10 cm
sin A =
12 cm
y
sin y =
o
h
10
12
sin y = 0.8333
y = sin-1 (0.8333)
y = 56.4o
Example: Given sec  = 4, find the values of the
other five trigonometric functions of  .
Solution:
Draw a right triangle with an angle  such
4
4
that 4 = sec  = hyp = .
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
15
θ
1
1 = 4
sin 
15
1
sec  =
=4
cos 
1
cot  =
15
csc  =
Applications Involving Right Triangles
A surveyor is standing 115 feet from the base of the
Washington Monument. The surveyor measures the angle
of elevation to the top of the monument as 78.3.
How tall is the Washington Monument?
Solution:
where x = 115 and y is the height of the monument. So, the height of the
Washington Monument is
y = x tan 78.3
 115(4.82882)  555 feet.
Fundamental Trigonometric Identities
Reciprocal Identities
sin  = 1/csc 
cot  = 1/tan 
cos  = 1/sec 
sec  = 1/cos 
tan  = 1/cot 
csc  = 1/sin 
Co function Identities
sin  = cos(90   )
sin  = cos (π/2  )
tan  = cot(90   )
tan  = cot (π/2  )
sec  = csc(90   )
sec  = csc (π/2  )
Quotient Identities
tan  = sin  /cos 
cos  = sin(90   )
cos  = sin (π/2  )
cot  = tan(90   )
cot  = tan (π/2  )
csc  = sec(90   )
csc  = sec (π/2  )
cot  = cos  /sin 
Pythagorean Identities
sin2  + cos2  = 1 tan2  + 1 = sec2  cot2  + 1 = csc2 