8-3 and 8-4 Trigonometry and Angles of Elevation and Depression

Quick Review

In right Δ A B C , C cap delta eh b c comma c is the right angle.

sin ∠ A = leg opposite ∠ A hypotenuse cos ∠ A = leg adjacent to ∠ A hypotenuse tan ∠ A = leg opposite ∠ A leg adjacent to  ∠ A table with 3 rows and 2 columns , row1 column 1 , sine angle eh , column 2 equals . fraction legopposite . angle eh , over hypotenuse end fraction , row2 column 1 , cosine angle eh , column 2 equals . fraction legadjacentto . angle eh , over hypotenuse end fraction , row3 column 1 , tangent angle eh , column 2 equals . fraction legopposite . angle eh , over legadjacentto . angle eh end fraction , end table

Example

What is FE to the nearest tenth?

You know the length of the hypotenuse, and F E ¯ f e bar is the side adjacent to ∠ E . angle e .

cos 41 ° = F E 9 Use cosine . F E = 9 ( cos 41 ° ) Multiply each side by  9. F E ≈ 6.8 Use a calculator . table with 3 rows and 3 columns , row1 column 1 , cosine , 41 degrees , column 2 equals , fraction f e , over 9 end fraction , column 3 cap usecosine . . , row2 column 1 , f e , column 2 equals 9 . open . cosine , 41 degrees . close , column 3 cap multiplyeachsideby . 9. , row3 column 1 , f e , column 2 almost equal to 6.8 , column 3 cap useacalculator . . , end table

Exercises

Express sin A , cos A , sine eh comma cosine eh comma and tan A tangent eh as ratios.

Find the value of x to the nearest tenth.

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18. While flying a kite, Linda lets out 45 ft of string and anchors it to the ground. She determines that the angle of elevation of the kite is 58 ° . 58 degrees . What is the height of the kite from the ground? Round to the nearest tenth.

8-5 Vectors

Quick Review

A vector is any quantity that has magnitude and direction. You can describe a vector by an ordered pair or by its magnitude and direction.

The sum of two vectors is the resultant. You can add vectors by adding their coordinates.

Example

What is the magnitude of the vector?

d = ( 75 − 0 ) 2 + ( − 150 − 0 ) 2   = 28125   ≈ 167.7050983 table with 3 rows and 2 columns , row1 column 1 , d , column 2 equals . square root of open , 75 minus 0 , close squared . plus . open . negative 150 minus 0 . close squared end root , row2 column 1 , , column 2 equals , square root of 28125 , row3 column 1 , , column 2 almost equal to . 167.7050983 , end table

The magnitude is about 168 miles.

Exercises

Find the magnitude and direction of each vector. Round to the nearest tenth.

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21. Find the sum of the vectors below. Express your answer as an ordered pair.

    

22. Navigation A whale-watching boat leaves port and travels 12 mi due north. Then the boat travels 5 mi due east. In what direction should the boat head to return directly to port?

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