Alg 3 Ch Angles in TRIGONOMETRY
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1 Alg Ch NOTES: 6. Angles in TRIGONOMETRY I Def. of radian: One radian is the measure of a central angle of a circle that is subtended by an arc whose length is equal to the radius of the circle. r Therefore arc length = angle in radians x radius rad r ex R R 0 The radius wraps itself around the circle times. Approx. 6.8 times.
2 Alg Ch x R 0, y SECTORS
3 Alg Ch GEOMETRY REVIEW RIGHT TRIANGLES 90 a 60 a a a a 0 a 60 0 SOHCAHTOA Trig Ratios for Right Triangles
4 Alg Ch Algebra Assignment # SKETCHPAD
5 Alg Ch Trigonometric Functions Intro: The unit circle is the circle with radius =, center is located at the origin. I Vocabulary A. Initial side: B. Terminal side: C. Coterminal angles: The initial and terminal sides form an angle at the center if the terminal side rotates CCW, the angle is positive if the termianl side rotates CW, the angle is negative unit circle positive negative coterminal RIGHT TRIANGLES Right Triangle Trigonometry II Trig or Circular Functions SOHCAHTOA θ theta is a variable used to represent an angle opp hyp θ adj θ
6 Alg Ch SOHCAHTOA A. sin θ = opp hyp adj B. cos θ = hyp C. tan θ = opp adj A A A B θ C B θ C B θ C A 6 θ 0 A θ C B C B B (0,) P(x,y) On the unit circle y sinθ = O x A (,0) cosθ = tanθ =
7 Alg Ch Triangles in the Unit Circle Signs of Trig functions Where functions are positive? I O II Reference Triangles Remember the special triangles w/ radius = A. Drop from point to x axis. 0 B. Examples. Find the sin
8 Alg Ch Find the sin 6. Find the 7 cos. Find the tan. Find the cos coterminal angles R = coterminal angles 6. Find the sin 0 = 7. Find the cos 6 = coterminal angles
9 Alg Ch Find sin, cos and tan. III Quadrangle Angles Def: An angle that has its terminal side on one of the coordinate axes. To find these angles, use the chart B (0,) y sinθ = = y x cosθ = = x y sin tanθ = = x cos C (,0) D (0, ) A (,0)
10 Alg Ch Complete the following tables. Algebra Assignment # Radian Degree Sin Cos Radian Degree Sin Cos
11 Alg Ch Algebra Assignment # Answers Radian Degree Sin Cos Radian Degree Sin Cos 0 0
12 Alg Ch Other Trigonometric Functions A. sinθ Cosecant: (csc) y cosθ tanθ Secant: (sec) Cotangent: (cot) x B. Find the following values. csc. sec. cot. 7 cot 6 C. Identities they come from the Pythagorean Triangle r= cos θ + sin θ = y divide by cos θ divide by sin θ θ x x + y =
13 Alg Ch DAY NOT ON THE UNIT CIRCLE. Find cosθ if sinθ = / and 0 θ. Find tanθ if sinθ = / 7 and θ. Find cscθ if cosθ = and θ. Find secθ if sinθ = / and θ. If Tan θ =, 70 < θ<60, find all the remaining functions of θ. 6. Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point (, ) on its termninal ray.
14 Alg Ch Algebra Assignment # Complete the following tables please. Radian 8 Degree Sin Cos Tan Cot Sec Csc Radian Degree Sin Cos Tan Cot Sec Csc
15 Radian 8 Degree Sin Cos Tan Cot Sec Csc Algebra Assignment # Answers Radian Degree Sin 0 Cos 0 0 Tan 0 Cot 0 Sec Csc 0
16 Algebra Review Worksheet Assignment # Remaining Trig Functions () Sin( θ ) =, 0 < θ <. Find the remaining trig. functions of θ. () Cos(θ ) =, < θ <. Find the remaining trig. functions of θ. () Tan( θ ) =, 80 < θ < 70. Find the remaining trig. functions of θ. () Sec(θ ) = 7, < θ <. Find the remaining trig. functions of θ. () Csc(θ ) =, < θ <. Find the remaining trig. functions of θ. (6) Cot(θ ) =, < θ < 0. Find the remaining trig. functions of θ. (7) Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point (, ) on its terminal ray. (8) Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point ( 0, ) on its terminal ray.
17 Algebra Review Worksheet Assignment # Answers () cos(θ ) =, tan( θ ) =, cot(θ ) =, sec(θ ) =, csc(θ ) = () sin( θ ) =, tan( θ ) =, cot(θ ) =, sec( θ ) =, csc(θ ) = () sin( θ ) =, cos(θ ) = 9, cot(θ ) = 9, sec( θ ) = 9, csc(θ ) = 9 () sin( θ ) = 0, cos(θ ) = 7 7, tan(θ ) = 0, cot(θ ) =, csc(θ ) = () sin( θ ) =, cos(θ ) =, tan(θ ) =, cot(θ ) =, sec( θ ) = (6) sin( θ ) =, cos(θ ) = 0 0, tan( θ ) =, sec(θ ) = 0, csc(θ ) = 0 (7) sin( θ ) =, cos(θ ) =, tan(θ ) =, cot(θ ) =, sec(θ ) =, csc(θ ) = (8) sin( θ ) =, cos(θ ) = 0, tan(θ ) is undefined, cot( θ ) = 0, sec( θ ) is undefined, csc(θ ) =
18 Algebra Review Worksheet Assignment # () Complete the following table please. Algebra Review Worksheet Radian 7 Degree Sin Cos Tan Cot Sec Csc () Cos(θ ) =, θ is in Quadrant II. Find the remaining trig. functions of θ. () Tan( θ ) =, θ is in Quadrant III. Find the remaining trig. functions of θ. () Csc(θ ) =, θ is in Quadrant IV. Find the remaining trig. functions of θ. () Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point (, ) on its terminal ray.
19 Alg () 9 Ch 6 Trig Answers () Radian 7 Degree Sin Cos Tan Cot Sec Csc () sin =, tan( θ ) =, cot( θ ) =, sec( θ ) =, csc( θ ) = () sin( θ ) =, cos( θ ) = 0, cot( θ ) =, sec( θ ) = 0 0, csc( θ ) = 0 () sin( θ ) =, cos( θ ) = 6, tan( θ ) =, cot( θ ) = 6, sec( θ ) = 6 6 () sin( θ ) =, cos(θ ) =, tan(θ ) =, cot(θ ) =, sec(θ ) =, csc(θ ) =
20 Alg () 0 Ch 6 Trig ADDITIONAL REVIEW. Convert the following to radians: a) b) 0 c) 700. Convert the following to degrees: a) 8 b) c). When angle is Θ is placed in standard position, its terminal side passes through the given point. Find the values for all six trig functions. a) (, ) b) (, ) c) (0, 7) d) (, ). Given the quadrant of φ and one of its six trig values, find the other five. a) sin φ =, φ in quadrant I b) tan φ =, φ in quadrant II c) sec φ = 7, tan φ < 0 d) cot φ =, sin φ < 0. Fill in the blanks for the following: a) r =, s =, θ =, A = b) r =, s =, θ =, A = c) r =, s = θ =, A = 0 d) r =, s =, θ =, A = 0 6. Find each of the following: a) sin b) cos c) sec 0 d) tan e) csc f) cot 7 g) cos 60 h) sin i) sin j) csc k) cos l) sec 6 7. Find cos (sin ). 8. Find cos (sin (cot ))
21 Alg () Ch 6 Trig Answers: 7. a) b) c) 0. a) 0 b) c) 0. a) b) sin θ = cos θ = tan θ = cot θ = sec θ = csc θ = sin θ = cos θ = tan θ = cot θ = sec θ = csc θ = c) sin θ = cos θ = 0 tan θ = und cot θ = 0 sec θ = und csc θ = d). a) b) c) d) sin θ = cos θ = tan θ = cot θ = sec θ = csc θ = cos θ = tan θ = cotθ = sec θ = cscθ = sin θ = cos θ = cot θ = sec θ = csc θ = cos θ = sin θ = tan θ = cot θ = csc θ = 7 7 cos θ = sin θ = tan θ = cot θ = sec θ = csc θ =. a), b), c) r =, s = 0 d) r = 0, θ = 0 6. a) b) c) d) e) dne f ) 0 g) h) i) j) k) l)
22 Alg () Ch 6 Trig 7. 8.
23 Alg () Ch 6 Trig Extra Review. Convert the following to radians a) 700 b) 00 c).. Convert the following to degrees: a) b) c) radian 80. When angle is Θ is placed in standard position, its terminal side passes through the point (, ). Find the values for all six trig functions.. When angle is Θ is placed in standard position, its terminal side passes through the point (6, 8). Find the values for all six trig functions.. Given that cos φ =, < φ<. Find the value of the other trig functions. 6. Given that cot φ = and csc φ <0. Find the value of the other trig functions. 7. If the arc length of a circle is 0 cm and the area of the sector it intercepts is 80 cm, find the radius of the circle and the angle of the sector. 8. Find each of the following a) sin b) cos 6 7 c) sec d) tan 800 e) csc f) cot 7 g) sin 0 h) sin i) cos 0 k) sec
24 Alg () Ch 6 Trig ANSWERS. a) 0 b) c) 900. a) b) c) 80. = = sin cos tan = cot = sec = csc =. sin = cos = tan = cot = sec = csc = 7 7. sin = tan = cot = sec = csc = sin = cos = tan = sec = csc = Radius: Angle: / 8. a) b) c) dne d) 0 e) 9. f) dne g) h) i) k)
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