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Trigonometry Basics
Right Triangle Trigonometry
Sine FunctionSine 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 functionSine function
Given a right triangle, and reference angle A:
sin A = hypotenuseopposite
A
oppositehypotenuse
The sin function specifies these two sides of the triangle, and they must be arranged as shown.
Sine FunctionSine 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 Note: If this did not work on your calculator, try
pressing the pressing the sinsin key first, then type-in 40. Press key first, then type-in 40. Press the = key to get the answer.the = key to get the answer.
Sine Function
Try each of these on your calculator: sin 55° sin 10° sin 87°
Sine FunctionSine Function
Sine Function
Try each of these on your calculator: sin 55° = 0.819 sin 10° = 0.174 sin 87° = 0.999
Sine FunctionSine Function
Inverse Sine FunctionInverse Sine Function
Using sin-1 (inverse sin):
If 0.7315 = sin θthen sin-1 (0.7315) = θ
Solve for θ if sin θ = 0.2419
Inverse Sine FunctionInverse Sine Function
Cosine function
The next trig function you need to know is the cosine function (cos):
cos A = hypotenuseadjacent
A
adjacent
hypotenuse
Cosine FunctionCosine Function
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 FunctionCosine Function
Cosine Function
Try these on your calculator: cos 25° cos 0° cos 90° cos 45°
Cosine FunctionCosine Function
Cosine Function
Try these on your calculator: cos 25° = 0.906 cos 0° = 1 cos 90° = 0 cos 45° = 0.707
Cosine FunctionCosine Function
Using cos-1 (inverse cosine):
If 0.9272 = cos θthen cos-1 (0.9272) = θ
Solve for θ if cos θ = 0.5150
Inverse Cosine FunctionInverse Cosine Function
Tangent function
The last trig function you need to know is the tangent function (tan):
tan A = adjacentopposite
A
adjacent
opposite
Tangent FunctionTangent Function
Tangent FunctionTangent 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 FunctionTangent Function
Tangent Function
Try these on your calculator: tan 5° = 0.087 tan 30° = 0.577 tan 80° = 5.671 tan 85° = 11.430
Tangent FunctionTangent Function
Using tan-1 (inverse tangent):
If 0.5543 = tan θthen tan-1 (0.5543) = θ
Solve for θ if tan θ = 28.64
Inverse Tangent FunctionInverse Tangent Function
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
adjoppA
hypadjA
hypoppA
tan
cos
sin
Review
Review
The sin function:
sin A = hypotenuseopposite
A
oppositehypotenuse
Review
The cosine function.
cos A = hypotenuseadjacent
A
adjacent
hypotenuse
Review
Review
The tangent function.
tan A = adjacentopposite
A
adjacent
opposite
Review
Most Common Application:
2 2
1
cossin
tan
r x yx ry r
yx
x
yr
θ
Review
Solve for x:x = sin 30°x = cos 45°x = tan 20°
Review
Review
Solve for θ:
0.7987 = sin θ0.9272 = cos θ2.145 = tan θ
Review
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,
.cos2
cos2
cos2
222
222
222
Cabbac
Baccab
Abccba
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
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