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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 will see some examples of these ratios.

Examples - 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.

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

Answer

We need to know r.

Using Pythagoras, we have

So

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

Answer

"Find all 6 trigonometric ratios of θ" means "find sin θ, cos θ, tan θ, csc θ, sec θ and cot θ".

First we need to find r:

---

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

45° - 45° Triangle

30° - 60° Triangle

Memory Aid

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

Exercises - Exact Values

Find exact values of the following:

1. sin 60°

2. cos 60°

3. tan 60°

4. csc 30°

5. cot 45°

6. sec 45°

Answers

Using the 30-60 triangle:

1.

2.

3.

4. csc 30° = 2

Q5 and Q6 need the 45-45 triangle:

5. cot 45° = 1

6.

 

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.]

Examples

Find using caclulator. Answer correct to 4 decimal places.

1) sin 49°

2) cos 27.53°

3) tan 26°35'57"

4) csc 18.34°

5) sec 5°34'72"

6) cot 73°

Answers

1) This is just one step on the calculator.

sin 49° = 0.7547

2) This question is also just one step on calculator, since it is in decimal degree form.

cos 27.53° = 0.8868

3) For this next one, you need to make sure that you know how to enter an angle in DMS form (degrees - minutes - seconds).

tan 26°35'57" = 0.5007

4) You do not have a "csc" button on your calculator, so you need to do this in 2 steps. Find the sine of 18.34° first, then press the "1/x" button (or "x^-1" button) to find the reciprocal.

csc 18.34° = 3.1781

5) Likewise with this one, you need to find cos 5°34'72" first, then press "1/x" button.

sec 5°34'72" = 1.0048

6) Similar to numbers 4) and 5), you need to find tan 73° first, then press "1/x" button.

cot 73° = 0.3057

 

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

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

Solution:

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^-10.3462 = 19.096°.

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

NOTE 1:

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.

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.

NOTE 2:

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 (strictly), 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:

 

Exercises 2 - Finding Angles

Find θ (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.)

Answers

1. This is straightforward - use the sin^-1 button on your calculator:

θ = sin^-10.6235 = 38.572°

2. This one uses the tan^-1 button:

θ = tan^-13.689 = 74.833°

Can you draw a triangle to illustrate what this means? Go on, try - it really helps to understand it.

3. We have to do some thinking for this one. There is no csc^-1 button on our calculators, so we need to proceed as follows.

csc θ = 8.32, so sin θ = 1/8.32 = 0.12019. (since sin θ is the reciprocal of csc θ).

Now we can use the sin^-1 button to obtain:

θ = sin^-10.12019 = 6.9032°

4. Similar to Q3, we need to find the reciprocal first.

sec θ = 6.96, so cos θ = 1/6.96 = 0.143678. So

θ = cos^-1 0.143678 = 81.739°

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Trigonometric Functions
• 1. Angles
• 2. Sine, Cosine, Tangent & Reciprocals
• 3. Values of Trigonometric Functions
• 4. The Right Triangle and Applications
• 5. Signs of the Trigonometric Functions
• 6. Trigonometric Functions of Any Angle
• 7. Radians
• 8. Applications of Radian Measure
• Pulleys
• 9. Radians and the Trigonometric Ratios

• Comments, Questions?

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