1. Find angle measurements in degrees and radians based on the unit circle. 1. Students understand the notion of angle and how to measure it, both in degrees and radians. They can convert between degrees and radians. Sketch: a) 210 on a unit circle b) -50 on a unit circle c) 520 on a unit circle d) π 4 2. Convert degrees to radians and radians to degrees. e) 7π 6 3 Convert to degrees: π radians. 4 Convert to radians: 210 π Express in degrees: radians 5 (**FW) 1 8 revolution Find the indicated angle B, if C is the center of the circle: C 5 B 8 1
3 1 2. Students know the definition of sine and 1. Know definition as coordinate point. Find θ. ( 2, 2) cosine as y and x coordinates of points on (x = l cos θ, y = 1 sin θ) the unit circle, and are familiar with the graphs of the sine and cosine functions. 2. Recognize the graphs of sine and cosine functions. Graph the functions f(x) = sin x and g(x) = cos x, where x is measured in radians, for x between 0 and 2π. Identify the points of intersection of the 2 graphs. Write an equation for the graph of this sine function. 2 1 π 2π Find an angle ß between 0 and 2π such that cos (ß) = cos (6π/7) and sin (ß) = - sin(6π/7). Find an angle θ between 0 and 2π such that sin (θ) = cos(6π/7) and cos (θ) = sin (6π/7). (**FW) Graph the functions f(x) = sin x and g(x) = cos x, where x is measured in radians, for x between 0 and 2π. Identify the points of intersection of the two graphs. (**FW) 2
3. Students know the identify cos 2 (x) + sin 2 1. Prove Pythagorean identity. Prove sin 2 θ + cos 2 θ = 1 (x) = 1. (cos θ, sin θ) unit circle 2. Apply Pythagorean identity. Simplify: sin 2 θ cos 2 θ - cos 2 θ Prove that sec 2 x + csc 2 x = sec 2 x csc 2 x. (**FW) 4. Students graph functions of the form f(t) = Asin (Bt + f) or f(t) = Acos (Bt +f), and interpret A, B, and f in terms of amplitude, frequency, period, and phase shift. 1. Find amplitude, period, phase shift, and frequency. 3π f(x) = 2 sin (3x + 4 ). amplitude = period = frequency = phase shift = 3π Graph f(x) = 2 sin (3x + 4 ) for -2π < x < 2π. 3
On a graphing calculator, graph the function f(x) = sin(x) cos(x). Select a window so that you can carefully examine the graph. a) What is the apparent period of this function? b) What is the apparent amplitude of this function? c) Use this information to express f as a simpler trigonometric function. (**FW) 5. Students know the definition of the tangent and cotangent functions, and can graph them. 1. Know definition of tangent and cotangent functions. sin x tan x = cos x If sin x = 1 2 in quadrant I, find tan x. cos x cot x = sin x = 1 tan x 2. Graph tangent and cotangent functions. Graph f(x) = tan x for [-2π, 2π]. 4
Match the equation to the graph. y = tan x y = tan x y = cot x y = cot x Use the definition of f(x) = tan(x) to determine the domain of f. (**FW) 6. Students know the definition of the secant and cosecant functions, and can graph them. 1. Know definitions of secant and cosecant functions. 1 1 sec x = cos x csc x = sin x 3 If sin x = in quadrant I, find sec x. 2 2. Graph secant and cosecant functions. Graph f(x) = sec x for [-2π, 2π]. Identify all vertical asymptotes to the graph of g(x) = sec x. (**FW) 5
7. Students know that the tangent of the 1. Know that tangent is the slope. Given tan θ = 3, find the equation of angle a line makes with the x-axis is equal 4 to the slope of the line. line m. θ (4,5) A line with positive slope makes an angle of 1 radian with the positive x- axis at the point (3,0). Find an exact equation for this line. (**FW) 8. Students know the definitions of the inverse trigonometric functions, and can graph the functions. 1. Know definition of inverse trigonometric functions. Match the following: 1. csc x a. sin -1 x 2. sin 1 x b. (sin x ) -1 3. arc sin x c. sin x -1 2. Graph inverse trigonometric functions. Graph y = arc sin x. π If tan(x) = tan( 5 ) and 3π < x < 4π, find x. (**FW) Graph f(x) = sin x and g(x) = sin -1 x on the same axes. In words, describe the relationship between the two graphs. (**FW) 6
1. Evaluate trigonometric functions in exact form. 9. Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. The terminal side of an angle θ in standard position contains the point (8,- 15). Find the value of all six trigonometric functions. Evaluate the expression. Assume the angle is in quadrant I. tan (sin -1 5 13 ). Find an angle α between 0 and -π for which cos (α) = 1 2. (**FW) 10. Students demonstrate understanding of the addition formulas for sines and cosines, their proofs, and use them to prove and/or simplify other trigonometric identities. 1. Understand sum and difference identities. Use the addition formula for sine to find the exact value for the function cos 75. Verify the following identity: sin (180 - θ) = sin θ. Use the addition formula for sine to find an expression for sin(75 ). (**FW) 7
11. Students demonstrate understanding of 1. Understand double and half-angle Use a half-angle identity to find the exact half-angle and double-angle formulas for sines and cosines, and can use them to identities. value for sin π 12. prove and/or simplify other trigonometric identities. If sin x = 3 and x is in quadrant I, find 5 the value of cos 2x. Verify the identity sin 2x = 2 cot x sin 2 x. Solve for θ, where 0 < θ < 2π : (cosθ)(sin2θ) - 2sinθ + 2 = 0. (**FW) 12. Students use trigonometry to determine unknown sides or angles in right triangles. 1. Use trigonometry to find sides and angles in right triangles. A tower 250 meters high casts a shadow 176 meters long. Find the angle of elevation of the sun to the nearest minute. Find the measure of the angle α in the triangle below. (**FW) 6 α 2 6 8
1. Know the Law of Sines and the Law of Cosines. 13. Students know the Laws of Sines and the Law of Cosines, and apply them to problems. A vertical pole sits between two points that are 60 feet apart. Guy wires to the top of that pole are staked at the two points. The guy wires are 40 feet and 35 feet long. How tall is the pole? (**FW) Solve for the distance c on the triangle below, if the angle A is 30. (**FW) 14. Students determine the area of a triangle given one angle and the two adjacent sides. 1. Find the area of a triangle. Find the area of the triangle below if the angle B measures 20. (**FW) 4 6 c 6 A B 4 15. Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates, and vice versa. 1. Plot polar coordinates. Find the polar coordinates of (-3,5). 9
2. Convert rectangular coordinates to polar. Convert polar coordinates to rectangular. 2 Graph (3, ) π Find all representations in polar coordinates of the point whose rectangular coordinates are (2 3, -2). (**FW) 16. Students represent equations given in rectangular coordinates in terms of polar coordinates. 1. Write rectangular equations. Write the rectangular equation x 2 + y 2 = 25 in polar form. Express the circle of radius 2 centered at (2,0) in polar coordinates. (**FW) 17. Students are familiar with complex numbers. They can represent a complex number in polar form, and know how to multiply complex numbers in their polar form. 1. Represent complete numbers in polar form. Express in polar form l + i. Use this to compute (i + 1) 3, (1 + i) 4 Represent i + 1 in polar form. Use this to compute (i + 1) 2. (**FW) 10
18. Students know De Moivre s Theorem, 1. Use De Moivre s Theorem. Find i. and can give n-th roots of a complex number given in polar form. Find all square roots of i. (**FW) 19. Students are adept at using trigonometry in a variety of applications and word problems. 1. Use trigonometry to solve word problems. The range of a projected object is the distance that it travels from the point where it is released. In the absence of air resistance, a projectile released at an angle of elevation, θ, with an initial velocity of V 0 has a range of R = sin θ, where g is the acceleration due to gravity. Find the range of a projectile with an initial velocity of 88 feet per second if sin θ = 3 5 and cos θ = 4 5. V 2 0 g The acceleration due to gravity is 32 feet per second. Find the area of a regular pentagon with a radius of 2. Express the answer to the nearest thousandth. 11
Two surveyors 560 yards apart sight a boundary marker C on the other side of a canyon at angles of 27 and 38. Their measurements will be used to plan a bridge that spans the canyon. How long will the bridge be, to the nearest tenth of a yard? Mr. Allen needs to rent a ladder to paint the outside of his house. The painted part of his house is 20 feet high, and the ladder must reach the top of the painted part. How long must the ladder be if the angle formed by the ladder and level ground is 75? (**FW) A person holds one end of a rope that runs through a pulley and has a weight attached to the other end. The section of rope between the person and the pulley is 20 feet long; the section of rope between the pulley and the weight is 10 feet long. The rope bends through an angle of 35 degrees in the pulley. How far is the person from the weight? (**FW) 12
How long does it take for a minute hand on a clock to pass through 1.5 radians? (**FW) A lighthouse stands 100 feet above the surface of the ocean. From what distance could it be seen? (You may assume that the radius of the earth is 3,960 miles.) (**FW) 13
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