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Right triangle trigonometry definitions. Build a right triangle in the diagram over the central angle. Label the right triangle. c is the Label the right triangle using theta( ) as the reference angle. This angle is always ACUTE! B c a HYPOTENUSE. OPPOSITE of . A The length Opposite of . The length of the Hypotenuse. a is the side b is thebside C ADJACENT to . a sin c The length Adjacent of . b cos The length of the Hypotenuse. c The length Opposite of . a tan The length Adjacent of . b SOH-CAH-TOA I P Y N P P E O O S T I E T N E U S E O S I N E D J A C E N T Y P O T E N U S E A N G E N T P P O S I T E D J A C E N T Right triangle trigonometry definitions. Build a right triangle in the diagram over the central angle. Label the right triangle. c is the Label the right triangle using theta( ) as the reference angle. This angle is always ACUTE! a2 b2 c2 B c HYPOTENUSE. Pythagorean Theorem a a is the side ADJACENT OPPOSITE of . A b b is the side The length Opposite of . The length of the Hypotenuse. REMEMBER C ADJACENT of . OPPOSITE ba sin c The length Adjacent of . ba cos The length of the Hypotenuse. c The length Opposite of . ba tan The length Adjacent of . ba SOH-CAH-TOA I P Y N P P E O O S T I E T N E U S E O S I N E D J A C E N T Y P O T E N U S A N G E N T P P O S I T E D J A C E N T Find the 6 trigonometric functions with respect to . Circle the reference angle and label the opposite, adjacent, and hypotenuse. Find the value of the missing side, c. a2 b2 c2 3 4 c 2 2 2 HYP. c 5 c 5 OPP. 9 16 c 2 25 c 2 SOH-CAH-TOA 3 sin 5 4 cos 5 5 csc 3 5 sec 4 3 tan 4 4 cot 3 ADJ. b Special Right Triangle Relationships. b2 b2 c2 2b c 2 2 c 2b 2 45 2 cb 2 b b 45 : 45 : 90 b:b:b 2 45 b 1:1: 2 1 2 sin 45 2 2 2 cos45 2 tan 45 1 c b 2 45 csc45 2 2 1 45 sec45 2 cot45 1 1 These answers are considered exact values. Special Right Triangle Relationships. a 2 b 2 2a 2 a 2 b 2 4a 2 b 3a 2 60 30 b 2 3a 2 ba 3 2a 2a ba 3 2 30 : 60 : 90 a : a 3 : 2a 1: 3 : 2 60 a a 2a 60 1 sin 30 2 csc30 2 30 3 cos30 2 2 2 3 sec30 3 3 1 3 tan 30 3 3 cot 30 3 2 3 60 1 3 sin 60 2 1 cos60 2 tan 60 3 csc 60 2 3 3 sec60 2 cot 60 3 3 Do you see a pattern? Cofunction Identities. Two positive angles are complimentary if their sum is 90o. Our trigonometric functions are identified with the prefix “Co”. Sine & Cosine Tangent & Cotangent Secant & Cosecant sin cos90 ac cos sin 90 bc tan cot90 ba cot tan 90 ba sec csc90 bc csc sec90 ac From the 30-60-90 Ex. 1 sin 30 2 1 cos90 30 cos60 2 cos sin 90 cos52 sin 90 52 cos52 sin 38 tan cot 90 tan 71 cot90 71 tan 71 cot19 sec csc90 sec24 csc90 24 sec24 csc66 sin cos90 sin 15 cos90 15 sin 15 cos75 csc90 10 csc2 20 90 10 2 20 80 2 20 80 3 20 60 3 20 Definition of Reference Angle. Let be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle formed by the terminal side of and the x – axis. Find all six trigonometric function exact values of 210. •Use special right triangles and reference angles to find the exact values of the trigonometric functions. 30 : 60 : 90 45 : 45 : 90 1: 3 : 2 1:1: 2 opp 1 SOH – CAH - TOA sin 210 hyp cos210 2 adj 3 hyp 2 tan 210 1 3 opp adj 3 3 cot 210 90 S A adj 210 180 3 30 1 opp 2 T Reference angle hyp 3 adj 3 1 opp hyp 2 2 3 sec210 adj 3 3 csc210 hyp 2 2 opp 1 C Use special right triangles and reference angles to find the exact values of the trigonometric functions. 30 : 60 : 90 45 : 45 : 90 1: 3 : 2 1:1: 2 SOH – CAH - TOA cos 240 S opp hyp 2 3 60 1 T A adj Reference angle tan 675 S 675 240 C 1 adj cos 240 hyp 2 T tan 675 A adj1 Reference angle 45 opp 2 hyp 1 C opp 1 1 adj 1 S opp A hyp 2 3 2 adj C 225 135 45 45 C S A adj 3 30 60 1 Reference angle A T opp 3 T adj 3 1 2 2 2 2 1 3 1 2 3 2 4 3 S A hyp 120 60 1 T S C 2 135 225 hyp T 1 3 2 opp 1 C 2 Give away for 45o angle MODE Right now we want DEGREE mode, move cursor to DEGREE and hit ENTER 2nd APPS activate the ANGLE window. x-1 button, Sine, Cosine, and Tangent Degree symbol. Minute symbol. are to the right. 0.7571217563 1 cos97.977 -7.205879213 tan 51.4283 -0.4067366431 sin 1 0.96770915 75.4 7524' 1.253948151 1 1.0545829 cos 1 1 Second symbol. cos 18.51470432 1830'53" 1.0545829 ALPHA, + F W sin 2500 sin 2.5 109 lb F W sin 5000sin 6.1 531 lb right acute C A B complimentary B hyp. Find angles first. 90 34.5 55.5 SOH-CAH-TOA a sin 34.5 12.7 a 12.7 sin 34.5 b cos34.5 12.7 b 12.7 cos34.5 c 12.7 A 34.5 b adj. B ____ 55.5 7.2 in a _____ 10.5 in b _____ opp. a C Need to find the following: hyp. opp. = 29.43 = 53.58 adj. A _____ 33.3 90 33.3 56.7 56 . 7 B _____ 44.77 b _____ A general rule is to always use the information you are given. We can find b by the Pythagorean Theorem. a b c 2 2 2 b c2 a2 b 53.582 29.432 Sides were given to 2 decimal places … so b = 44.77 SOH-CAH-TOA 29.43 sin A 53.58 29.43 A sin 1 53.58 Make sure the MODE is DEGREE. SOH-CAH-TOA hyp. 25 tan 20 x 25 x tan 20 x x tan 20 25 x tan 20 tan 20 x 25 tan 20 y =opp. 57.635 40 x =adj. h 68.687 20 adj. 25 opp. hyp. y tan 40 68.687 68.687 tan 40 y h 25 57.635 h 82.635 ft When a single angle is given, it is understood that the Bearing is measured in a clockwise direction from due north. N N N 45o 165o 225o Starts at a Bearing starting on the north-south line and uses an acute angle to show the direction, either east or west. S 45o E N 75o W N 45o S 75o C adj. x 61o 29 opp. 29 61 29 A 61 3.7mi SOH-CAH-TOA 331o B hyp. x cos29 3.7 x 3.7 cos29 3.7 3.7 x AC 3.2mi C 47 43o 3.5(22) = 77 4(22) = 88 47o P 43 c s2 p2 c2 77 2 882 c 2 S c 772 882 117 nautical miles h 27 hyp. hyp. adj. SOH-CAH-TOA h tan 15 x 50 x 50 tan 15 h x 50 x 50 h x 50 tan 15 opp. adj. Solve for x to find h. h tan 28 x h x tan 28 h x tan 28 x x 50 tan 15 x tan 28 tan 15 x 50 tan15 x tan28 x tan 15 50 tan 15 x tan 28 x tan 15 x tan 15 50 tan 15 x tan 28 x tan 15 50 tan 15 xtan 28 tan 15 tan 28 tan 15 tan 28 tan 15