Unit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.

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1 Name Unit #17: Spring Trig Unit Notes #1: Basic Trig Review I. Unit Circle A circle with center point and radius. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount. B. If you know the first quadrant and the quadrantals, you really know the entire circle. (see handout) II. Reference Angles A. A reference angle must have a measure between and (in degrees) or and (in radians). B. To find the reference angle, you find the measure of the angle formed by the ray and the -axis. C. By quadrants: III. Inverses A. Notation for inverses: or B. Inverses have restricted domains Each trigonometric inverse function exists in only 2 quadrants. They all exist in quadrant and then they must also have a negative quadrant that quadrant I. Quadrants that the inverse trig. functions exist: I and II :,, IV and I:,, C. Using a Calculator to find Inverses i. If you have arcsin, arcos, or arctan, just hit [trig function] then the number. Make sure you match the mode of the question. ii. If the inverses are the reciprocal functions:,, or then you must hit [ trig]( 1 number ) ii. If the number is negative, you must input a positive into the calculator and then use your reference angles to move it to the correct quadrant.

2 IV. Solving A. Notation: B. Determine where the trig. function is equal to the given fraction on the. C. Is the fraction or. Which quadrants is this trig. function or in? D. Rotate to both quadrants using. E. Using a calculator: 1. If it is sin/ cos/ or tan, hit [trig] and the number. 2. If it is csc/ sec/ or cot, hit [trig] and the number. 3. If the number is negative, you must input a positive into the calculator and then use your reference angles to move it to the correct quadrant. IV. Model Problems Guided Practice Evaluate: sin 6 π On Your Own Evaluate: cos - 6 π Find the reference angle when given a rotation of 328 o Find the reference angle when given a rotation of 143 o Find the reference angle when given a rotation of 7π 8 Find the reference angle when given a rotation of 10π 13 Evaluate: 1 arcsin 2 Evaluate: 1 arccos 2 arctan ( 1) Evaluate: Evaluate: arccot 3

3 3 cos arccos 2 Evaluate 1 ( ) Evaluate sin tan ( 1) Evaluate 1 5 cos csc 3 Evaluate tan 1 5π tan 4 Use a calculator to evaluate arctan( 0.45) Use a calculator to evaluate 1 sin (0.487) Use a calculator to evaluate 1 sec (4.25) csc( 8.81) Use a calculator to evaluate arc Solve 2cos x 1= 0 Solve 2sin x + 1= sec x = Solve 3 Solve cot = 3

4 Solve cos x = sin x = Solve 9 Solve 8 cot x = Solve 13 csc x = Notes #2: Solve and Trig Identities I. Identities A. Quotient 1. tan θ = 2. cot θ = B. Reciprocal Identities 1. sin θ = 2. csc θ = 3. cos θ = 4. sec θ = 5. tan θ = 6. cot θ =

5 C. Pythagorean Identities sin θ cos θ 1 + = This also means: and 2 2 tan θ + 1 = sec θ This also means: and 2 2 cot θ + 1 = csc θ This also means: and D. Sum/Difference and Double-Angle Formulas 1. Sum/Difference: sin( x + y) = cos( x + y) = tan( x + y) = 2. Double - Angles sin 2x = cos 2x = or = or = tan 2x = Solve tan 2x = 1 Given: cos =, < < and sin = 0< <, find sin, 24 3π Given: cos x =, π < x <, find 25 2 sin (2x) cos (2x)

6 tan (2x) Solve 6.sec tan =3 Notes #3: Solving Triangles I. Law of Sines A. Not every triangle is a. Therefore we can not always use unit circles or basic trigonometric functions. B. Law of Sines allows us to solve - triangles by making proportions. sin sin sin C. Law of Sines: A = B = C. a b c D. You can use Law of Sines anytime you have an and it s side. E. Cross-multiply and solve. Make sure you write your equation in calculator-ready form.

7 F. Other tidbits i. You can not use this law to find the angle. II. Law of Cosines ii. Any time you want to find an angle, you must hit sin. iii. Lower case letters are while capital letters are. A. You may not have an angle AND it s opposite side so you would not be able to use of. B. Law of Cosines CAN find an angle and so if you are looking for 3 angles, find the first using the Law of. C. Recall that you must hit cos to find an. D. Formulas Associated with Law of Cosines i c = a + b 2ab cos C ii. The sides can be any letter and so c is really the side you are looking for when you already know and and also angle. You must know the angle opposite the side you are looking for. a + b c iii. If you are looking for an angle, you can use: cos C = 2ab iv. The side opposite the you are looking for must be behind the sign in the formula. III. Right Triangles A. If given 2 sides you can use Pythagorean Theorem:. B. If given 1 side and 1 acute angle you use. IV. Model Problems Solve the triangle if a = 10, b = 22, and c = 31.

8 Solve the triangle if B = 28.7 o, C = o, and b = Notes #4: Graphing Polar Polar Graphing I. Introduction A. Polar Coordinate System Instead of the point (x, y) it will now be (, ) with r = and = angle of rotation. B. Pole The fixed point (like the ) C. Polar Axis The initial ray (like the - ) D. To plot a point: rotate. 1. Rotate the angle first. If it is positive then rotate - but if it is negative, 2. From this location, move the radius. The radius is how many you move out. 3. If r is negative, follow the opposite ray across the pole. GP OYO π Plot 4, 3 5π Plot 10, 6 5π Plot 1, 4 7π Plot 2, 6

9 II. How to Convert A. Graph the given point B. If it is rectangular form (x, y) and you are to convert to polar ( r, θ ) 1. Find r: r = 2. Find θ: θ = 3. If it is from the unit circle, you must use radians without decimals. C. If it is polar form and you are to convert to rectangular form 1. From the unit circle, write the coordinate that matches the rotation 2. the point by the radius. 3. x = and y = III. Model Problems Convert ( 2,π ) to rectangular π Convert 3, to rectangular 6 Convert ( 1,1 ) to polar Convert (0, 2) to polar

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11 Notes #5: Vectors I. Vectors A. Definition: Quantities that have and B. Initial Point: point (P) C. point the final point (Q D. Magnitude the length of the vector ( PQ) Found just like the NORM. E. Component of a vector - a, b II. Operations Using Mathematics A. Scalar Multiplication Multiply the vector by a quantity a. B. Addition Add the x components and add the y components C. Subtraction the negative then add. III. Graphing Vectors A. Vectors can move anywhere on the plane. You must only keep the (length) and (angle measure) B. When adding put the tail of the 2 nd vector on the of the first vector. Connect the original to the new. IV. Unit Vectors C. To subtract, go in the direction of the 2 nd vector A. Unit Vector a vector with length (magnitude) 1 B: Find a unit vector 1. Find the length of the vector 2. Divide each component by the length of the vector. C. A unit vector is in the same direction as the original vector V. Alternate Vector Notation linear combination of i and j A. i = <1,0> and j = <0,1> B. v = < a, b > = ai + bj C. This form is called the resultant vector (v = ai + bj) D. The sum of all the forces acting on an object is called the resultant force. VI. Components of the Direction Angle A. If v = =, then a = and b = where is the direction angle of v. B. The angle =

12 VII. Model Problems Guided Practice Ex1: Given u and v, find u + v, u v and 2u + 3v. u v On Your Own Ex2: Find the component vector and magnitude if P = (-2, 6) and Q = (4,-3) Ex3: Let u = 5,2 and v = 3,1, find 4u 3v and 4 3 Ex4: Find the unit vector in the direction of = 5,12 Ex7: Find the resultant vector if P = (-2, -4) and Q = (-4, -7) Ex8: u = 2i 6j and v = -5i + 2j Find u+v and 2u - v

13 Ex9: Find the direction angle of each vector u = 5i + 13j and v = -10i + 7j Ex 10: Find the component form of the vector that represents the velocity of an air plane at the instant its wheels leave the ground, if the plan is going 60 miles per hour and the body of the plan makes a 7 angle with the horizontal. Ex 11: An object at the origin is acted upon by two forces. A 150-pound force makes an angle of 20 with the positive x-axis, and the other force of 100-pounds makes an angle of 70 with the positive x-axis. Find the direction and magnitude of the resultant force. Ex 12: An airplane is flying due south with an air speed of 300 miles per hour. There is a 35 mph wind from the direction 20. Find the course and ground speed of the plane.

14 Notes #5 Part 2 I. Operation with vectors A. Dot Product: If =, and =, then = B: Cross Product: If =, and =, then = II. Properties of Dot Product If u, v, and w are vectors, and k is a real number, then: A. = B. = C. = D. = E. 0 = II. Angle between vectors A. If is the angle between the nonzero vectors u and v, cos _ = III. Parallel, perpendicular (orthogonal) or neither C. Vectors u and v are parallel when =0 D. Vectors u and v are orthogonal when =0 IV. Model Problems Ex 1: Given = 5,3 = 2,6 Find and Ex 2: Given =4 2 = 3 Find and Ex 3: Find the angle between the vectors = 3,1 = 5,2 Ex 4: Find the angle between the vectors =4 3 = 2

15 Ex 5: Given = 1,3, = 1,2 and = 2, 5 Find Ex 6: Given = 1,3, = 1,2 and = 2, 5 Find Ex 7: Determine whether vectors u and v are parallel, orthogonal or neither. = 2, 6 = 9,3 Ex 8: Determine whether vectors u and v are parallel, orthogonal or neither. =3 6 = 5 3 Notes #6: Parametric equations I. Definition of a plane curve Let f and g be continuous functions of f on an interval. The set of all points (x,y), where x = f(t) and y = g(t) is called a plane curve. The variable t is called a parameter and the equations that define x and y are called parametric equations. II. III. Graphing plane curves a. Make a chart b. Plot points c. Sketch curve Eliminating the Parameter a. Solve one of the equations for t b. Plug t into the other equation c. Simplify the result to eliminate the parameter.

16 IV. Model Problems Ex 1: Graph the curve by hand = 2 =4 4, 1 3 Ex 2: Graph the curve by hand = 4 =, 2 3 Ex 3: Eliminate the parameter = 2 =4 4, 1 3 Ex 4: Eliminate the parameter = 4 = 2, 2 3 Ex 5: Eliminate the parameter =5cos =5sin, 0 2 Ex 6: Eliminate the parameter =2cos =5sin, 0 2