Trigonometry Basics
Right Triangle Trigonometry
Sine Function

When you talk about the sin of an angle,
that means you are working with the
opposite side, and the hypotenuse of a
right triangle.
Sine function

Given a right triangle, and reference angle A:
sin A =
opposite
hypotenuse
The sin function specifies
these two sides of the
triangle, and they must be
arranged as shown.
hypotenuse
opposite
A
Sine Function
For example to evaluate sin 40°…
 Type-in 40 on your calculator (make sure
the calculator is in degree mode), then
press the sin key.
 It should show a result of 0.642787…

 Note:
If this did not work on your calculator,
try pressing the sin key first, then type-in 40.
Press the = key to get the answer.
Sine Function
Try each of these on your calculator:
 sin 55°
 sin 10°
 sin 87°

Sine
Function
Sine
Function
Try each of these on your calculator:
 sin 55° = 0.819
 sin 10° = 0.174
 sin 87° = 0.999

Inverse Sine Function

Using sin-1 (inverse sin):
If
then

0.7315 =
sin-1 (0.7315) =
Solve for θ if sin θ = 0.2419
sin θ
θ
Cosine
function
Cosine
Function

The next trig function you need to know
is the cosine function (cos):
cos A =
adjacent
hypotenuse
hypotenuse
A
adjacent
Cosine Function
Use your calculator to determine cos 50°
 First, type-in 50…
 …then press the cos key.
 You should get an answer of 0.642787...

 Note:
If this did not work on your calculator,
try pressing the cos key first, then type-in 50.
Press the = key to get the answer.
Cosine Function
Try these on your calculator:
 cos 25°
 cos 0°
 cos 90°
 cos 45°

Cosine
Function
Cosine
Function
Try these on your calculator:
 cos 25° = 0.906
 cos 0° = 1
 cos 90° = 0
 cos 45° = 0.707

Inverse Cosine Function

Using cos-1 (inverse cosine):
If
then

0.9272 =
cos-1 (0.9272) =
Solve for θ if cos θ = 0.5150
cos θ
θ
Tangent Function
function

The last trig function you need to know
is the tangent function (tan):
opposite
adjacent
tan A =
opposite
A
adjacent
Tangent Function
Use your calculator to determine tan 40°
 First, type-in 40…
 …then press the tan key.
 You should get an answer of 0.839...

 Note:
If this did not work on your
calculator, try pressing the tan key first,
then type-in 40. Press the = key to get the
answer.
Tangent Function
Try these on your calculator:
 tan 5°
 tan 30°
 tan 80°
 tan 85°

Tangent Function
Try these on your calculator:
 tan 5° = 0.087
 tan 30° = 0.577
 tan 80° = 5.671
 tan 85° = 11.430

Inverse Tangent Function

Using tan-1 (inverse tangent):
If
then

0.5543 =
tan-1 (0.5543) =
Solve for θ if tan θ = 28.64
tan θ
θ
Review
Review
These are the only trig functions you will
be using in this course.
 You need to memorize each one.
 Use the memory device: SOH CAH TOA

opp
hyp
adj
cos A 
hyp
opp
t an A 
adj
sin A 
Review

The sin function:
sin A =
opposite
hypotenuse
hypotenuse
opposite
A
Review
Review

The cosine function.
cos A =
adjacent
hypotenuse
hypotenuse
A
adjacent
Review
Review

The tangent function.
opposite
adjacent
tan A =
opposite
A
adjacent
Most Common Application:
r x y
2
2
x  r cos 
y  r sin 
 y
  tan  
 x
1
r
θ
x
y
Review
Review
Solve for x:
x = sin 30°
x = cos 45°
x = tan 20°

Review
Review

Solve for θ:
0.7987 = sin θ
0.9272 = cos θ
2.145 = tan θ
What if it’s not a right triangle?
- Use the Law of Cosines:
The Law of Cosines
In any triangle ABC, with sides a, b, and c,
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C.
What if it’s not a right triangle?

Law of Cosines - The square of the magnitude
of the resultant vector is equal to the sum of the
magnitude of the squares of the two vectors, minus two
times the product of the magnitudes of the vectors,
multiplied by the cosine of the angle between them.
R2 = A2 + B2 – 2AB cosθ
θ