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Trigonometry

Trigonometry

Trigonometry can be used to help us find the angles and distances of triangles. In GCSE Mathematics, you will have come across sine, cosine and tangent - functions that come from the angles and distances of a right-angled triangle. Three other functions are the reciprocals of these familiar functions. They are secant (sec), cosecant (cosec) and cotangent (cot) respectively.

Using trigonometric functions

You can use trigonometric functions to help you find the angles and lengths of sides in triangles. The first step is to label your triangle with opposite, adjacent and hypotenuse; the diagram below shows you how to label your triangle correctly.

Trigonometry right-angled triangle StudySmarterLabeling a right-angled triangle, Thomas-Gay - StudySmarter Original

SOHCAHTOA

SOHCAHTOA is an acronym to help you remember which function you need to use when using trigonometry to find an angle or the length of a side in a triangle. They are broken down for you below:

Trigonometry SOHCAHTOA Study SmarterSOHCAHTOA, Thomas-Gay - StudySmarter Original

These equations help you to find the missing length of a side in a right-angled triangle. Here are the steps:

  1. Label your triangle with hypotenuse, opposite, adjacent, as shown in the diagram above.

  2. Choose the equation that you need to use based on the information you have from the triangle.

  3. Substitute your figures into the equation

  4. Use your calculator and the sin, cos, tan buttons to find your missing length.

If you need to find the missing angle in your triangle, you can still use SOHCAHTOA but with the inverse of the sin, cos and tan functions. Here are the equations for finding the missing angle:


Find the value of x

Trigonometry right-angled triangle StudySmarterWorked example of labeling a triangle, Thomas-Gay - StudySmarter Original

To do this, we label our triangle, with hypotenuse, opposite and adjacent, shown below.

Trigonometry right-angled triangle StudySmarterWorked example of labeling a triangle, Thomas-Gay - StudySmarter Original

You can see that you have got the value of the opposite side, and you are looking for your hypotenuse.

Using SOHCAHTOA, your calculation involves the O and the H. The part of SOHCAHTOA that includes both of these letters is SOH. Therefore you need to use sine to find the value of x;

Now you can substitute your values into the formula and rearrange it to solve for x;

Non-right-angled triangles

Trigonometry can be used to find the angles and side lengths for non-right-angled triangles too. To do this, you use the sine and cosine rules.

When using these rules, you first need to label the triangle to help you see which function to use.


Trigonometry non-right-angled triangle StudySmarterLabeling a non-right-angled triangle, Thomas-Gay - StudySmarter Original

When labeling the triangle, it doesn't matter which angles are which, as long as the sides are correctly matched to their opposite angle.

Cosine rule

You can use the cosine rule to find the length of a missing side when you know the other two sides and the angle between them. The formula to use is:

You can also use the cosine rule to find an angle if you know all of the lengths of the sides. Here, the formula is:


Sine rule

You can use the sine rule to find the length of a side or the angle of a triangle. The formula used to find the side length is below:

The formula for a missing angle is:

Let's have a look at an example and decide which rule would be best to use.

Find the value x

Trigonometry non-right-angled triangle StudySmarterWorked example to find the correct formula for a triangle, Thomas-Gay - StudySmarter Original

Looking at this example, you can see that the missing quantity x is at an angle. You have also been given the side lengths for all three sides, and because of this, you can use the cosine rule - specifically, the formula used to find an angle.

Area of a triangle

If you know the lengths of two sides of any triangle and the angle between them, you can find the area of the triangle. The formula for this is:


Let's look at this triangle and think about how we put the information into our formula.

Trigonometry area of ​​a triangle StudySmarterArea of a triangle, Thomas-Gay - StudySmarter Original

First, you need to label your triangle, and then you can substitute the information into the formula;

What is a unit circle?

A unit circle can help you understand the trigonometric functions, the circle has a radius of 1, and the center coordinates are (0,0). This means that when you calculate the trigonometric values, you will get a coordinate point on the circle, known as (x, y).

Trigonometry unit circle StudySmarterUnit example, Thomas-Gay - StudySmarter Original

To find the gradient of p, you use

Trigonometry - Key takeaways

Frequently Asked Questions about Trigonometry

Trigonometry is a mathematical topic which looks at the relationship between angles and distances.

Trigonometry was invented in Greece. The word comes from trigon, which means triangle and metron, which means to measure.

To find an angle using trigonometry, you use the inverse function of either sin, cos or tan, also known as sin-1, cos-1 and tan-1.

Final Trigonometry Quiz

Question

What are the three trigonometric functions?

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Answer

Sine, cosine and tangent

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Question

What are the names of the three sides of a right-angled triangle?

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Answer

Hypotenuse, opposite and adjacent

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Question

What does ‘CAH’ represent?

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Answer

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Question

How do you find the length of a side in a non-right-angled triangle?

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Answer

You can use the sine or cosine rule.

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Question

What is the formula for the cosine rule?

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Answer

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Question

What formula would you use when finding the side length, when you are given the hypotenuse and the opposite side?

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Answer

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Question

What formula would you use when finding a missing angle, when you are given the adjacent and opposite sides?

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Answer

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Question

In a unit circle, what does the x coordinate of p represent

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Answer

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Question

In a unit circle, what does the y coordinate of p represent?

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Answer

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Question

When would you use the sine rule?

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Answer

You would use the sine rule when finding the missing side length or angle when you have two opposite sides and angles.

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Question

What is the equation for tan?

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Answer

opposite/ adjacent

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Question

What is the equation for secant?

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Answer

Hypotenuse divided by adjacent.

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Question

What are the reciprocal functions?

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Answer

arcsine

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Question

Which function would I use to find the angle that has a sine value of \(0.73\)?

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Answer

The arcsin function.

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Question

What is the amplitude of a tangent graph?

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Answer

None of these.

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Question

What is the sine value of \(x=\frac{\pi}{6}\)?

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Answer

\(\dfrac{1}{2}\).

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Question

What is the cosine value of \(x=\dfrac{\pi}{3}\)?

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Answer

\(\dfrac{1}{2}\).

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Question

What is the tangent value of \(\dfrac{\pi}{2}\)?

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Answer

Undetermined.

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Question

What happens to the tangent graph at \(x=-\dfrac{\pi}{2}\)?

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Answer

There is a vertical asymptote.


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