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Trigonometry Lecture Notes_part2

# Cos 2 2 cos cos 2cos cos 2 2 cos cos 2sin sin 2 2 = =

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Unformatted text preview: cos 2 2 cos cos 2cos cos 2 2 cos cos 2sin sin 2 2 +- + =- +- = +- + = +-- = - Example 71 Express each sum or difference as a product: a. sin9 sin5 x x + b. cos4 cos3 x x- Example 72 Express the difference as a product: sin 4 sin 2 x x- Example 73 Verify the following identity: sin sin tan cot sin sin 2 2 x y x y x y x y + +- =- Example 74 Verify the following identity: cos3 cos5 tan sin3 sin5 x x x x x- = + Section 7.2 Trigonometric Equations This section involves equations that have a trigonometric expression with a variable, such as cos x . To understand this section we must consider a simple equation such as cos x = 0.5. Example 75 Example 76 Equations with a multiple angle Solve the equation: tan3 1 x = , within the interval:0 2 x &amp;lt; Example 77 Equations with a multiple angle 3 sin 2 2 x = , 2 x &amp;lt; Example 78 Example 79 2 tan sin 3tan x x x = , within 0 2 x &amp;lt; Example 80 Using a trigonometric identity to solve a trig equation: cos 2 3sin 2 x x +- = , within 0 2 x &amp;lt; Example 81 Using a trigonometric identity to solve a trig equation: 1 sin cos 2 x x = , within 0 2 x &amp;lt; Example 82 Using a trigonometric identity to solve a trig equation: sin cos 1 x x- = , within 0 2 x &amp;lt; Example 83 Solve the following equation: 7cos 9 2cos + = - Example 84 Solve the equation on the interval [0,2 ): Example 85 Solve the equation on the interval [0,2 ) Example 86 Solve the equation on the interval [0,2 ) Section 6.7 Applied Problems Solving a right triangle means finding the missing lengths of the sides and the measurements of its angles. We will label the right triangle as is done in the following diagram: Example 87 3 tan 2 3 = sin 2 sin x x = 2 cos 2cos 3 x x +- = Finally, we need to find c. Because we have a known angle, a known adjacent side, and an unknown hypotenuse, we use the cosine function. cos34.5 = 10.5/c c=10.5/cos34.5 = 12.74 In summary, B = 55.5, a = 7.22, and c = 12.74. Example 88 Example 89 A 200 foot cliff drops vertically into the ocean. If the angle of elevation of a ship to the top of the cliff is 22.3 degrees, how far above shore is the ship? Example 90 A building that is 250 feet high cast a shadow that is 40 feet long. Find the angle of elevation of the sun at that time. Example 91 A boat leaves the entrance to a harbor and travels 40 miles on a bearing S 64 E. How many miles south and how many miles east from the harbor has the boat traveled?...
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cos 2 2 cos cos 2cos cos 2 2 cos cos 2sin sin 2 2 = = = =...

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