4.12.1 Trigonometry

The student is able to (I can):

For any right triangle

• Define the sine, cosine, and tangent ratios and their inverses

• Find the measure of a side given a side and an angle

• Find the measure of an angle given two sides

• Use trig ratios to solve problems

By the Angle-Angle Similarity Theorem, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure. This means that the ratios between the sides of those triangles are always the same.

Because these ratios are so useful, they were given names: sinesinesinesine, cosinecosinecosinecosine, and tangenttangenttangenttangent. These ratios are used in the study of trigonometry.

sine

cosine

tangent

sine of ∠A

cosine of ∠A

tangent of ∠A

AAAA

hypotenuse

adjacent

opposite

leg opposite AsinA

hypotenuse

∠= =

leg adjacent to AcosA

hypotenuse

∠= =

leg opposite AtanA

leg adjacent to A

∠= =

∠

We can use the trig ratios to find either missing sides or missing angles of right triangles. To do this, we will set up equations and solve for the missing part. In order to figure out the sine, cosine, and tangent ratios, we can use either a calculator or a trig table.

To use the Nspire calculator to find tan 51°:

• From a New Document, press the µ key:

• Use the right arrow key (¢) to select tan and press ·:

• Type 5I and hit ·:

To use the calculator on your phone:

• Turn your phone landscape to access the scientific calculator.

• Depending on your phone, you will either either either either ttttype the angle in first and select tan, ororororselect tan and then type in the angle.

To find an angle, we use the inverseinverseinverseinverse trig functions (in more advanced classes, you will hear them referred to as arcsine, arccosine, and arctangent). On your calculator, these are listed as sin—1, cos—1, and tan—1.

Ex. Find :

Press the µ button, and then the ¤ arrow to select sin—1. Then enter 8p17·. You should get 28.07…

This means that the angle opposite a leg of 8 with a hypotenuse of 17 will measure around 28˚.

−

1 8sin

17

You will be expected to memorize these ratio relationships. There are many hints out there to help you keep them straight. The most common is SOHSOHSOHSOH----CAHCAHCAHCAH----TOATOATOATOA , where

A mnemonic I like is “Some Old Hippie Caught Another Hippie Trippin’ On Acid.”

Or “Silly Old Hitler Couldn’t Advance His Troops Over Africa.”

pOS

pin

pHy=

dAC

jos

pHy=

pOT

pan

jAd=

Examples I. Use the triangle to find the following ratios.

1. sin A = _____

2. cos A = _____

3. tan A = _____

A

BC

8

15

17

Examples I. Use the triangle to find the following ratios.

1. sin A = _____

2. cos A = _____

3. tan A = _____

A

BC

8

15

17

8

17

15

17

15

8

Examples I. Use the triangle to find the following ratios.

4. sin B = _____

5. cos B = _____

6. tan B = _____

A

BC

8

15

17

8

17

15

17

8

15

Examples II. Find the lengths of the sides to the nearest tenth.

1.

2.

x (opp)

15(adj)

58°

26

(hyp)

x(adj)

46°

° =

= °

≈

xtan58

15x 15tan58

24.0

° =

= °

≈

xcos46

26x 26cos46

18.1

III. Find the missing angle to the nearest whole degree.

26 (hyp)

19 (opp)

xº

° =19

sinx26

− =

1 19x sin

26

≈ °x 47