3. Values of the Trigonometric Functions

by M. Bourne

In the last section, Sine, Cosine, Tangent and the Reciprocal Ratios, we learned how the trigonometric ratios were defined, and how we can use x-, y-, and r-values (r is found using Pythagoras' Theorem) to evaluate the ratios.

Now we'll see some examples of these ratios.

Finding Exact Values of Trigonometric Ratios

Find the exact values indicated. What this means is don't use your calculator to find the value (which will normally be a decimal approximation). Keep everything in terms of surds (square roots). You will need to use Pythagoras' Theorem.

Example 1

Find the exact value of sin θ if the terminal side of θ passes through `(7, 4)`.

Continues below

Example 2

Find the exact values of all 6 trigonometric ratios of θ if the terminal side of θ passes through `(2, 10)`.

The following two cases are very common in the study of exact trigonometric ratios.

45° - 45° Triangle

45-45 degrees triangle

`sin\ 45^text(o)=text(opp)/text(hyp)=1/sqrt2`

`cos\ 45^text(o)=text(adj)/text(hyp)=1/sqrt2`

`tan\ 45^text(o)=text(opp)/text(adj)=1/1=1`

30° - 60° Triangle

30-60 degrees triangle

`sin\ 30^text(o)=text(opp)/text(hyp)=1/2`

`cos\ 30^text(o)=text(adj)/text(hyp)=sqrt3/2`

`tan\ 30^text(o)=text(opp)/text(adj)=1/sqrt3`

Memory Aid

In the `30-60` triangle, it is easy to forget where to put the `1, 2, sqrt(3)` sides and the angles. You could remember it like this:

Take an equilateral triangle, sides 2 units:

Equilateral triangle with sides 2 units

Now, cut it in half horizontally:

Equilateral triangle cut in two

Take the top half only. The unknown side is the `sqrt(3)` and the `30^@` and `60^@` angles are as indicated:

30-60 trangle showing sides 2, 1, and root 3

Example 3 - Exact Values

Find exact values of the following:

a. sin 60°

b. cos 60°

c. tan 60°

d. csc 30°

e. cot 45°

f. sec 45°

Finding Trigonometric Ratios Using Calculator

Suggestion: Go find the instruction book for your calculator. You are sure to need it in this section. Each calculator brand and model is a bit different - please don't expect your teacher to know how to use every model of every brand of calculator!

Caution: Make sure your calculator is set correctly to degree mode (not radian mode!) for this section. [We learn about radians later. It is very easy to mess up these problems when you are mixing degrees and radians - always check that your answer is reasonable before moving on.]

Example 4

Find using caclulator. Answer correct to 4 decimal places.

a. sin 49°

b. cos 27.53°

c. tan 26°35'57"

d. csc 18.34°

e. sec 5°34'72"

f. cot 73°

Finding Angles Given The Trig Ratio

We are now going to work the other way around. We may know the final trigonometeric ratio, but we don't know the original angle.

Example 5

Find θ, given that tan θ = 0.3462 and that 0° ≤ θ < 90°.


We need to use the inverse tangent function (not the reciprocal function, as we did for cot θ). Our answer will be an angle. So we use the "`tan^-1`" button on our calculator, and we have:

θ = tan-1 0.3462 = 19.096°.

Check: We can use our calculator to check our answer: tan 19.096° = 0.3462. Checks OK.


It is very common (and better) to use `"arctan"` instead of "`tan^-1`". You will often see "`arctan`" throughout this site, rather than tan-1. It helps us to remember the difference between the reciprocal ratio (`cot`) and the inverse function (`arctan`).

In the above example, we would write:

θ = arctan 0.3462 = 19.096°.

You'll also see "`arcsin`", "`arccos`", "`"arccsc"`" etc. See more on this in Trigonometry Functions of Any Angle.


Be very careful with the difference between (eg) "`sin^-1`" and "`csc`". They are NOT the same!

Example: sin-1 0.935 = 69.23° (this gives us an angle).

But csc 0.935 = 1.2429 (there is no degree sign on `0.935`, so it must be in radians).

This is the `csc` of the angle `0.935` radians. It is a ratio, not an angle, and as you can see, it has a different value. We meet radians later in 7. Radians.

For the record, `csc\ 0.935` means:

`csc\ 0.935=1/(sin\ 0.935)=1.2429` (in radians)

Exercises - Finding Angles

Find θ for 0° ≤ θ < 90°, given that

1. sin θ = 0.6235

2. tan θ = 3.689

3. csc θ = 8.32

4. sec θ = 6.96

(I have restricted the domain for θ from `0°` to `90°` because we haven't seen how to solve it for angles greater than `90°` yet.)


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