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Radians

Radians

Degrees are the well-known angle measurement that can be used in trigonometry. An alternative is the concept of radians.

What are radians?

Radians are an alternative angle unit that make more sense if we consider a circle – so let's do that.

STEP 1: Let's start by drawing a circle with a unit radius (radius of 1).

Radians Circle, Radians, StudySmarterA general unit circle

A general unit circle

STEP 2: Let's start moving this radius by angle.

Radians, Radius of a circle moved by an angle, StudySmarterA general unit circle with a sector of angle

STEP 3: Now let's draw in a side to make this a right angled triangle. Label this side x.

Radians a right angled triangle in a circle, StudySmarterA triangle with opposite side x

STEP 4: We can use trigonometry now to write a calculation for x:

STEP 5: Now we can think about what would happen if the radius travelled the entire way around the circle. The radius would travel a distance of . This is because the circumference of the circle is .

Therefore radians

That's our key conversion:

How do we solve trigonometric equations using radians?

This uses exactly the same trigonometric methods as we would when using degrees. You can adjust a calculator to display answers in radians:

Option -> Angle Unit -> Radians

Find the value of angle CAB, give your answer in radians.

Radians, Finding angles, StudySmarter

SOLUTION:

Standard trigonometry question, figure out which of SOHCAHTOA we're using.

We have opposite and adjacent so TOA.

So we have to just perform an inverse trigonometric function so

Remember to make sure your calculator is in radians before performing this last step.

So Angle CAB has a size of .

Let's look at another example like this.

Find the value of .

Radians, Finding angles, StudySmarter

SOLUTION:

We can use the sine rule here therefore

Make sure your calculator is correctly set in radians. This means that:

We have seen how the calculator can be converted into radians form. Now let's look at how we can convert directly between radians and degrees.

Converting between radians and degrees

There is a key conversion:

And vice versa:

Let's see this in action.

Covert into radians.

SOLUTION:

And let's look at an example in the reverse direction.

Covert to degrees.

SOLUTION:

In the table below we can see some standard conversions between degrees and radians:

DegreesRadians

Radians - Key takeaways

    • Radians are an angle unit like degrees but with a small conversion.

    • Remember that π radians = 180 °.

    • When solving trigonometry problems or equations the method is exactly the same with radians and degrees.

    • Remember to set the angle unit to radians on your calculator. This will simplify the answers to problems.

Frequently Asked Questions about Radians

A radian is an angle unit like degrees, to help simplify solving trigonometric equations.

There are 2π radians in a circle.

Simplify divide by 180, then multiply your answer by π.

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of the users don't pass the Radians quiz! Will you pass the quiz?

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