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TRIGONOMETRY

TRIGONOMETRY. INVERSE TRIGONOMETRC RATIOS. . iNTRODUCTION. Triangles are the simplest polygons, having only three sides. Right triangles have exactly one interior angle measures 90 degrees. The Sum of the measures of the interior angles of a triangle is 180 degrees.

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TRIGONOMETRY

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  1. TRIGONOMETRY INVERSE TRIGONOMETRC RATIOS.

  2. iNTRODUCTION • Triangles are the simplest polygons, having only three sides. Right triangles have exactly one interior angle measures 90 degrees. • The Sum of the measures of the interior angles of a triangle is 180 degrees. • Special Right Triangles are triangles with specific angle measures and special relationships among the hypotenuse and the legs. Their measures are 45-45-90 and 30-60-90 • Trigonometry means triangle measurement. The ratio of two sides of a right triangle determine the measures of the acute angles. This measure is expressed in fractions of degrees converted to decimals. • To remember these ratios you can think SOH-CAH-TOA: Sine = Opposite over Hypotenuse Cosine = Adjacent over Hypotenuse Tangent = Opposite over Adjacent

  3. SUMMARY: RIGHT TRIANGLES A B C Two given sides One side length One side length + + + one missing side 45 degree angle One angle Pythagorean Theorem Special Right Triangle TRIGONOMETRY or or One side length two sides length + and 30 or 60 degrees angle a missing angle measure Special Right Triangle TRIGONOMETRY

  4. For each triangle determine the value of the unknown side or angle as represented with a variable. a) b) c) x 16 in 6 m y x 130 ft. 12 in d) y 26m 18m 52 ˚ y˚

  5. INVERSE TRIGONOMETRIC RATIOS In exercise d If you know the sin, cos or tan of an acute angle of a right triangle you can use an inverse trigonometric ratio to find the measure of that angle. The inverse trigonometric ratio will give you the measure of the angle when you know the ratio of two sides. Notation: The inverse trigonometric ratio for sine: The inverse trigonometric ratio for cosine: The inverse trigonometric ratio for tangent: In your calculator: - - number - = or Shift - - number - =

  6. Angle of ELEVATION AND ANGLE OF DEPRESSION If you are looking up, then the angle that you line of sight makes with a horizontal line is the angle of elevation. If you are looking down, then the angle that you line of sight makes with the horizontal is the angle of depression. Angle of elevation Horizontal Angle of depression

  7. EXAMPLE There are building codes that limit the angle of elevation of a set of steps to prevent accidents. The angle must be less than Find the measure of the angle corresponding to this set of stairs. What do you have? What are you looking for? - the run of each stair = 10 in - the measure of the angle x - the rise of each stair = 12in What can you add? - the height of the triangle - the base of the triangle So, h = 40 in and b = 48in Now your ready to select the trigonometric ratio. Use SOH – CAH – TOA. tan x = tan x = 0.83 Apply the inverse ratio: 0.83 = 39.8 - The measure of the angle is 39.8˚ less than 45˚ 10in 12 in x˚

  8. Example 2 Find the missing height using the diagram. What do you have? What are you looking for? What can you add? - two angles - the height of the triangle - the height is a perpendicular segment - the hypotenuse length - the little triangles are special right - a special right triangle triangles In the large triangle the length of the side opposite to 30˚ is 25 and the length of the side opposite to 60˚ is 25 In the little special triangle at the left : In the little special triangle at the right: h is the shorter leg h is the longer leg the hypotenuse is 25 the hypotenuse is 25 25 h h h 30˚ 60˚ 50 m

  9. Sin 30˚= or sin 60˚ = Sin 30 * 25 = h sin 60˚ * 25 = 21.6 = h 21.65 = h The height of the triangle is 21.65

  10. Problem Solving Tips • Read the problem slowly. • Study the diagram or create a diagram. • Look for definitions or properties you can use in the solution. • Try to explain the problem to yourself or partner. • REMEMBER • The best problem solvers makes lots of mistakes; but part of what make them so good is that they • DON’T GIVE UP.

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