Trignometry vbei

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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.


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 Function

Try each of these on your calculator: sin 55° = 0.819 sin 10° = 0.174 sin 87° = 0.999


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 Function

Try these on your calculator: cos 25° cos 0° cos 90° cos 45°


Cosine Function

Try these on your calculator: cos 25° = 0.906 cos 0° = 1 cos 90° = 0 cos 45° = 0.707


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 Function

Try these on your calculator: tan 5° = 0.087 tan 30° = 0.577 tan 80° = 5.671 tan 85° = 11.430


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

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The sin function:

sin A = hypotenuseopposite

A

oppositehypotenuse

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The cosine function.

cos A = hypotenuseadjacent

A

adjacent

hypotenuse

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The tangent function.

tan A = adjacentopposite

A

adjacent

opposite

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Most Common Application:

2 2

1

cossin

tan

r x yx ry r

yx

x

yr

θ

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Solve for x:x = sin 30°x = cos 45°x = tan 20°

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Solve for θ:

0.7987 = sin θ0.9272 = cos θ2.145 = tan θ

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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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