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 = hypotenuse

opposite

A

opposite

hypotenuse

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. key first, then type-in 40. Press the = key to get the answer.Press 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 θ

then 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 = hypotenuse

adjacent

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 θ

then 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 = adjacent

opposite

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 θ

then 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

adj

oppA

hyp

adjA

hyp

oppA

tan

cos

sin

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The sin function:

sin A = hypotenuse

opposite

A

opposite

hypotenuse

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The cosine function.

cos A = hypotenuse

adjacent

A

adjacent

hypotenuse

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The tangent function.

tan A = adjacent

opposite

A

adjacent

opposite

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Most Common Application:

2 2

1

cos

sin

tan

r x y

x r

y r

y

x

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 θ

0.9272 = cos θ

2.145 = 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