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Applications of Cosine

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Determine and Use the Cosine Ratio

Do you know how to use cosines when problem solving? Take a look at this dilemma.

A triangle has a hypotenuse of 4.5 inches. Angle A is equal to 40 degrees. Find the length of the adjacent side.

To figure this out, you will need to know how to use angle measures and cosines. You will learn how to accomplish this task in this Concept.

Guidance

Let's think about trigonometric ratios.

A trigonometric ratio for a specific angle will remain constant no matter how large or small the triangle is. The idea is that the sides will always be in proportion to each other. So, if you know the measure of an angle (and can therefore identify the value of a trigonometric ratio) and the value of one side, you can use trigonometry to calculate the lengths of other sides.

The trick is to use good algebra technique, and make sure that every time you set up a ratio, you are putting the values and variables in the correct places.

You can find trigonometric ratios by using your calculator.

You understand trigonometric ratios and have had a chance to practice reading specific values out of a table.

You can find the ratio for any trigonometric value using your calculator. Take a moment to locate the buttons for sine, cosine, and tangent on the calculator. Keep in mind that usually, sine is abbreviated as sin , cosine is usually abbreviated as cos , and tangent is usually abbreviated as tan.

Press the key of the ratio you want to find, and enter the angle in question. If you hit enter, or calculate, the calculator will show you the value of that specific ratio.

Let's look at finding the cosine ratio by using a calculator.

\text{cosine} \  23^{\circ}

You can find the values for each ratio using your calculator. When dealing with large decimals values, it is usually best to round the numbers to the nearest thousandth. It gives you a reasonably accurate value without being too long of a number to work with.

The cosine of 23^{\circ} is 0.92050485345244..., or about 0.921.

Now as we work with cosines, we will be using given information to find the length of the adjacent side of a right triangle.

Remember that the adjacent side is the side next to the angle that we are working with. As you recall, the ratio of cosine is \frac{adjacent}{hypotenuse} .

If you know the cosine value of the angle in question, and the length of the hypotenuse, you can find the measure of the adjacent side.

Look at the algebraic situation below.

\text{cosine} \angle X&=\frac{adjacent}{hypotenuse}\\\text{cosine} \angle X \times hypotenuse&=\frac{adjacent}{hypotenuse} \times hypotenuse\\\text{cosine} \angle X \times hypotenuse&=adjacent

If you multiply the cosine of any angle X and the length of the hypotenuse, the result is the length of the adjacent side.

Write this statement in your notebook. Be sure to include that it is for cosines.

Take a look at this situation.

What is the length of side BC in the triangle below?

Use the following equation to find the length of the side adjacent to angle B . Notice that to find the length of the adjacent side that you will first need to find the cosine for angle B . Then you can multiply that answer with the length of the hypotenuse. This will give you the measurement of the side next to or adjacent to the angle.

\text{cosine} \angle B \times hypotenuse&=adjacent\\\text{cosine} 14.5 \times 5&=adjacent\\0.968 \times 5 &=adjacent\\4.84 &=adjacent

The length of side BC is 4.84 units.

Use a calculator to find each cosine. You may round to the nearest hundredth.

Example A

Cosine 45^{\circ}

Solution:  .71

Example B

Cosine 62^{\circ}

Solution:  .47

Example C

Cosine 22^{\circ}

Solution:  .93

Now let's go back to the dilemma from the beginning of the Concept.

To work through this dilemma, we can use the following equation and solve.

\text{cosine} \angle A \times hypotenuse&=adjacent\\\text{cosine} 40 \times 4.5&=adjacent\\.77 \times 4.5 &=adjacent\\3.46 &=adjacent

The missing length of the adjacent side is 3.46 inches.

Vocabulary

Sine
a ratio between the opposite side and the hypotenuse of a given angle.
Cosine
a ratio between the adjacent side and the hypotenuse of a given angle.
Tangent
a ratio between the opposite side and the adjacent side of a given angle.
Trigonometric Ratio
used to find missing side lengths of right triangles when angle measures have been given.

Guided Practice

Here is one for you to try on your own.

A triangle has a hypotenuse of 7.5 inches. Angle A is equal to 55 degrees. Find the length of the adjacent side.

Solution

To do this, we can use the following equation.

\text{cosine} \angle A \times hypotenuse&=adjacent\\\text{cosine} 55 \times 7.5&=adjacent\\.57 \times 7.5 &=adjacent\\4.27 &=adjacent

The length of the adjacent side is 4.27 inches.

Video Review

Khan Academy Trigonometry Word Problems

Practice

Directions: Use a calculator to find each cosine. You may round to the nearest hundredth.

1. \text{Cosine} \  33^{\circ}

2. \text{Cosine} \  29^{\circ}

3. \text{Cosine} \  73^{\circ}

4. \text{Cosine} \  88^{\circ}

5. \text{Cosine} \  50^{\circ}

6. \text{Cosine} \  67^{\circ}

7. \text{Cosine} \  42^{\circ}

8. \text{Cosine} \  18^{\circ}

9. \text{Cosine} \  9^{\circ}

Directions: Find the length of the adjacent side.

10. A triangle has a hypotenuse of 7 inches. Angle A is equal to 60 degrees. Find the length of the adjacent side.

11. A triangle has a hypotenuse of 12 inches. Angle B is equal to 45 degrees. Find the length of the adjacent side.

12. A triangle has a hypotenuse of 8 inches. Angle A is equal to 35 degrees. Find the length of the adjacent side.

13. A triangle has a hypotenuse of 12 inches. Angle A is equal to 28 degrees. Find the length of the adjacent side.

14. A triangle has a hypotenuse of 6 inches. Angle A is equal to 33 degrees. Find the length of the adjacent side.

15. A triangle has a hypotenuse of 14 inches. Angle A is equal to 72 degrees. Find the length of the adjacent side.

16. A triangle has a hypotenuse of 11 inches. Angle A is equal to 80 degrees. Find the length of the adjacent side.

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