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Trigonometric Ratios with a Calculator

Solving for values when triangles aren't special right triangles.

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Trigonometric Ratios with a Calculator

What if you wanted to find the missing sides of a right triangle with angles of 20 and 70 and a hypotenuse length of 10 inches? How could you use trigonometry to help you? After completing this Concept, you'll be able to solve problems like this one.

Watch This

CK-12 Foundation: Chapter8TrigonometricRatioswithaCalculatorA

James Sousa: Determining Trigonometric Function Values on the Calculator

Guidance

The trigonometric ratios are not dependent on the exact side lengths, but the angles. There is one fixed value for every angle, from 0 to 90. Your scientific (or graphing) calculator knows the values of the sine, cosine and tangent of all of these angles. Depending on your calculator, you should have [SIN], [COS], and [TAN] buttons. Use these to find the sine, cosine, and tangent of any acute angle. One application of the trigonometric ratios is to use them to find the missing sides of a right triangle. All you need is one angle, other than the right angle, and one side.

Example A

Find the trigonometric value, using your calculator. Round to 4 decimal places.

a) sin78

b) cos60

c) tan15

Depending on your calculator, you enter the degree and then press the trig button or the other way around. Also, make sure the mode of your calculator is in DEGREES.

a) sin78=0.97815

b) cos60=0.5

c) tan15=0.26795

Example B

Find the value of each variable. Round your answer to the nearest tenth.

We are given the hypotenuse. Use sine to find b, and cosine to find a. Use your calculator to evaluate the sine and cosine of the angles.

sin2230sin22b=b30=b11.2  cos22=a3030cos22=a   a27.8

Example C

Find the value of each variable. Round your answer to the nearest tenth.

We are given the adjacent leg to 42. To find c, use cosine and use tangent to find d.

cos42ccos42c=adjacenthypotenuse=9c=9=9cos4212.1tan42=oppositeadjacent=d99tan42=d  d27.0

Any time you use trigonometric ratios, use only the information that you are given. This will result in the most accurate answers.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter8TrigonometricRatioswithaCalculatorB

Concept Problem Revisited

Use trigonometric ratios to find the missing sides. Round to the nearest tenth.

Find the length of a and b using sine or cosine ratios:

cos20=a1010cos20=aa9.4sin20=b1010sin20=bb3.4sin70=a1010sin70=aa9.4cos70=b1010cos70=bb3.4

Vocabulary

Trigonometry is the study of the relationships between the sides and angles of right triangles. The legs are called adjacent or opposite depending on which acute angle is being used. The three trigonometric (or trig) ratios are sine, cosine, and tangent.

Guided Practice

1. What is tan45?

2. Find the length of the missing sides and round your answers to the nearest tenth: .

3. Find the length of the missing sides and round your answers to the nearest tenth: .

Answers:

1. Using your calculator, you should find that tan45=1?

2. Use tangent for x and cosine for y.

tan2811tan28x=x11=x5.8  cos28=11y11cos28=y   y12.5

3. Use tangent for y and cosine for x.

tan4016tan40y=y16=y13.4  cos40=16x16cos40=x   x20.9

Practice

Use your calculator to find the value of each trig function below. Round to four decimal places.

  1. sin24
  2. cos45
  3. tan88
  4. sin43
  5. tan12
  6. cos79
  7. sin82

Find the length of the missing sides. Round your answers to the nearest tenth.

  1. Find sin80 and cos10.
  2. Use your knowledge of where the trigonometric ratios come from to explain your result to the previous question.
  3. Generalize your result to the previous two questions. If sinθ=x, then cos?=x.
  4. How are tanθ and tan(90θ) related? Explain.

Vocabulary

Hypotenuse

Hypotenuse

The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle

Legs of a Right Triangle

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Trigonometric Ratios

Trigonometric Ratios

Ratios that help us to understand the relationships between sides and angles of right triangles.

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