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Three Trigonometric Ratios

10.1. Three Trigonometric Ratios. Today we will… determine sines, cosines, and tangents of angles estimate or determine exact values of sine and cosine ratios define sine, cosine, and tangent use all three ratios to determine unknown lengths in real life. TANGENT.

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Three Trigonometric Ratios

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  1. 10.1 Three Trigonometric Ratios

  2. Today we will… • determine sines, cosines, and tangents of angles • estimate or determine exact values of sine and cosine ratios • define sine, cosine, and tangent • use all three ratios to determine unknown lengths in real life

  3. TANGENT The tangent of an angle is the ratio of the opposite leg with the adjacent leg. opp. leg hypotenuse opp. leg tan θ = adj. leg θ adj. leg

  4. There are two other ratios that can be used to find missing sides or angles. cosine cos AND sine sin

  5. COSINE The cosine of an angle is the ratio of the adjacent leg to the hypotenuse. adj. leg hypotenuse opp. leg cos θ = hyp. θ adj. leg

  6. Sine The sine of an angle is the ratio of the opposite leg and the hypotenuse. opp. leg hypotenuse opp. leg sin θ = hyp. θ adj. leg

  7. How do we remember these???

  8. REMEMBER… SOH – CAH- TOA opp. sin θ = hyp. adj. cos θ = opp. hyp. tan θ = adj.

  9. Let’s look at some examples… H 15 8 G U 17 8 15 Find… sin U = sin G = 17 17 15 8 cos G = cos U = 17 17 Notice which values are equal.

  10. Values of sine, cosine, and tangent can be estimated by drawing pictures. Estimate cos 54° B Draw a right triangle with a 54° angle and then measure the sides. 3.6 cm 2.9 cm 54° C A 2.1 cm 2.1 Calculator: cos 54= .588 cos 54° = = .583 3.6

  11. A supporting wire 8 feet long is to extend from the top of a pole 6 feet high. What is the angle between wire and the pole? We can use these ratios to estimate missing values in triangles. 6 cos θ= θ 8 8 ft. 6 6 ft. cos-1 θ= 8 41.4° θ=

  12. How far up on the side of a building can a 15 m ladder reach if the measure of the angle it makes with the ground may not exceed 72°? x 15 • sin 72°= • 15 15 x 15 x = 15 • sin 72° 72° x = 14.3 meters

  13. There are some exact values that you MUST MEMORIZE. All of these can be derived from the special right triangles.

  14. Try the next few on your own… A ladder leans against a house so that the base of the ladder is 6 feet away from the edge of the house. If the angle between the ladder and the ground is 60°, a. How long is the ladder? 12 feet b. How far up the side of the house is the ladder? 10.4 feet

  15. I want to know how tall the light post is. I am standing 8 feet from the post. I am 5’6” and looking up at the pole at a 50° angle. How tall is the pole? x 50 8 feet= 96 in. 66 in. x x = 96 tan 50 = 114.41 tan 50 = 96 114.41 + 66= 180.41 in. or about 15 ft

  16. An architect wants to build another overhang similar to the one at the right. However, he forgot to measure the angle that the overhang makes with the house. He knows the measurements shown. Use them to find the missing angle. θ 7 ft 4.5 ft 4.5 sin-1 = 40 ° θ = 7

  17. Sources • Microsoft PowerPoint (for the AutoShapes and animation sequences) • UCSMP Algebra II book • Microsoft Clip Gallery, sound j0082197.mid

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