Skip to main content

9. Radians and the Trigonometric Ratios

In this section we see examples of how to use radians instead of degrees when finding the values of sin, cos, tan, csc, sec and cot of angles.

We are using all that we have learned in this chapter. If you are lost:

Examples Involving the Trig Functions

Find the value of

1. `cos(pi/6)`

2. `sec 4.5`

3. `sec 4.5^"o"`


Using calculator, we find:

(1) `cos(pi/6) = cos 30^"o" =\ 0.8660254`

30o - 60o triangle showing `cos(pi/6) = 0.8660254`.

(2) Note there is no degrees sign, so this is radians. Using our calculator in radian mode, we get:

`sec 4.5 = -4.7439275`

The angle 4.5 radians.

(3) This time there is a degree sign, so on changing our calculator mode to degrees, we get:

`sec 4.5^"o" = 1.0030922`


The angle 4.5 degrees.

Notice the difference between these last two. Without the degree sign, 4.5 means "4.5 radians". It is important to set your calculator properly before starting these problems.

Please support IntMath!

Try the following exercises.

Continues below


1. Find `tan((4pi)/3)`


Using our calculator, we obtain:


For reference, `(4pi)/3 = (4xx180)/3=240^"o"`.

Please support IntMath!

2. Find `sin 2.34`


Using our calculator, we obtain:

`sin 2.34 = 0.7185`

Always check your calculator is in radians mode if the angle is in radians!

3. Find `θ` if `cos θ = -0.9135` `(0 ≤ θ < 2π)`


`cos θ = -0.9135` (`0 ≤ θ < 2π`)

We first find the relevant acute angle by solving the positive case, `cos α = 0.9135` (which is what we were doing in the "Reference Angle" section in 6. Trigonometric Functions of Any Angle).

We obtain:

`α = 0.41899` radians

Since `cos θ` is negative, it means θ is in the second and third quadrants.


`θ = π - 0.41899 = 2.723`, or

`θ = π + 0.41899 = 3.561`

4. Find `θ` if `csc θ = 3.940\ (0 ≤ θ < 2π)`


If `csc θ = 3.940`, then `sin theta=1/3.940=0.25381`.

We first find the relevant acute angle.

`α = 0.2566`.

Now θ is in the first and second quadrants because `csc θ` is positive.


`θ = 0.2566` (first quadrant), or

`θ = π - 0.2566 = 2.8850` (second quadrant)

Get the Daily Math Tweet!
IntMath on Twitter


Search IntMath, blog and Forum

Search IntMath

Online Trigonometry Solver

This trigonometry solver can solve a wide range of math problems.

Trigonometry Lessons on DVD

Math videos by

Easy to understand trigonometry lessons on DVD. See samples before you commit.

More info: Trigonometry videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

See the Interactive Mathematics spam guarantee.