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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.

Try the following exercises.


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


Using our calculator, we obtain:


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

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)

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