CHAPTERS 5-7 TRIG. FORMULAS PACKET

Transcription

1 CHAPTERS 5-7 TRIG. FORMULAS PACKET

2 PRE-CALCULUS SECTION 5-2 IDENTITIES Reciprocal Identities sin x = ( 1 / csc x ) csc x = ( 1 / sin x ) cos x = ( 1 / sec x ) sec x = ( 1 / cos x ) tan x = ( 1 / cot x ) cot x = ( 1 / tan x ) Quotient Identities tan x = ( sin x / cos x ) cot x = ( cos x / sin x ) Pythagorean Identities sin 2 x + cos 2 x = 1 or sin 2 x = 1 cos 2 x or cos 2 x = 1 sin 2 x 1+ tan 2 x = sec 2 x or 1 = sec 2 x tan 2 x or tan 2 x = sec 2 x 1 1+ cot 2 x = csc 2 x or 1 = csc 2 x cot 2 x or cot 2 x = csc 2 x 1 θ Cofunction Identities ( If is in radians, replace 90 o with /2. ) sin = cos ( 90 o ) csc = sec ( 90 o ) cos = sin ( 90 o ) sec = csc ( 90 o ) tan = cot ( 90 o ) cot = tan ( 90 o ) π

3 PRE-CALCULUS SECTION 5-5 IDENTITIES Graphing Variations of the y = A sin (Bx - C) + D and the y = A cos (Bx - C) + D functions. 1. Write the function in the form of y = A sin (Bx - C) + D or y = A cos (Bx - C) + D filling in the appropriate values for any constants A, B, C, or D not already listed in the original function. 2. Identify the amplitude using the formula - amplitude = I A I. 3. Identify the period using the formula - period = 2 π / B. 4. Identify the phase shift, if any, using the formula - phase shift = C/B. 5. Identify the vertical shift, if any, using the formula y = D. 6. Find (period / 4 ) which is the value which you will add to each x value to make your table. 7. Make a Table: Start with the value of x 1 where x 1 = phase shift. To find each subsequent value for x add the value solved for in step 6 to the previous x. (For example: x 2 = x 1 + (period / 4 ), x 3 = x 2 + (period / 4 ), etc.). 8. Find the corresponding values of y for the five key points by evaluating the function at each of the values of x from step Connect the five key points with a smooth curve and graph one complete cycle of the function. 10. Extend the graph in step 9 to the left or right as desired.

4 PRE-CALCULUS SECTION 5-6 IDENTITIES Graphing Variations of the y = A tan (Bx - C) + D function. 1. Write the function in the form of y = A tan (Bx - C) + D filling in the appropriate values for any constants A, B, C, or D not already listed in the original function. 2. Find two consecutive asymptotes by finding an interval containing one period using: Bx - C = - ( π / 2) call it Asymptote 1 and Bx - C = ( π / 2) call it Asymptote 2 3. Identify an x-intercept midway between the two consecutive asymptotes using: midpoint 1 = ( Asymptote 1 + Asymptote 2 ) / 2 4. Find the two points on the graph ( 1 / 4 ) and ( 3 / 4 ) of the way between the consecutive asymptotes using: ( 1 / 4 ) midpoint 1 = ( Asymptote 1 + x-intercept 1 ) / 2 and ( 3 / 4 ) midpoint 2 = ( Asymptote 2 + x-intercept 1 ) / 2 5. Make a table using the following values for: x x 1 = x value at Asymptote 1 y y 1 = undefined - which creates an Asymptote x 2 = ( 1 / 4 ) midpoint 1 y 2 = - A + D x 3 = x-midpoint 1 y 3 = D x 4 = ( 3 / 4 ) midpoint 2 y 4 = A + D x 5 = x value at Asymptote 2 y 5 = undefined - which creates an Asymptote 6. Connect the five key points with a smooth curve and graph one complete cycle of the function. 7. Extend the graph in step 6 to the left or right as desired.

5 Graphing Variations of the y = A cot (Bx - C) + D function. 1. Write the function in the form of y = A cot (Bx - C) + D filling in the appropriate values for any constants A, B, C, or D not already listed in the original function. 2. Find two consecutive asymptotes by finding an interval containing one period using: Bx - C = 0 call it Asymptote1 and Bx - C = π call it Asymptote2 3. Identify an x-intercept midway between the two consecutive asymptotes using: midpoint 1 = ( Asymptote 1 + Asymptote 2 ) / 2 4. Find the two points on the graph ( 1 / 4 ) and ( 3 / 4 ) of the way between the consecutive asymptotes using: ( 1 / 4 ) midpoint 1 = ( Asymptote 1 + x-intercept 1 ) / 2 and ( 3 / 4 ) midpoint 2 = ( Asymptote 2 + x-intercept 1 ) / 2 5. Make a table using the following values for: x y x 1 = x value at Asymptote1 y 1 = undefined - which creates an Asymptote x 2 = ( 1 / 4 ) midpoint 1 y 2 = A + D x 3 = x-midpoint 1 y 3 = D x 4 = ( 3 / 4 ) midpoint 2 x 5 = x value at Asymptote 2 y 4 = - A + D y 5 = undefined - which creates an Asymptote 6. Connect the five key points with a smooth curve and graph one complete cycle of the function. 7. Extend the graph in step 6 to the left or right as desired.

6 PRE-CALCULUS SECTION 6-1 IDENTITIES Guidelines for verifying Trigonometric Identities 1. Work with each side of the equation independently of the other side. Start with the more complicated side and transform it in a step-by-step fashion until it looks exactly like the other side. 2. Analyze the identity and look for opportunities to apply the fundamental identities. 3. Try using one or more of the following techniques: A. Rewrite the more complicated side in terms of sine and cosine. B. Factor out the greatest common factor. C. Separate a single-term quotient into two terms: a + b = a + b or a - b = a - b c c c c c c D. Combine fractional expressions using the least common denominator. E. Multiply the numerator and the denominator by a binomial factor that appears on the other side of the identity. 4. Don t be afraid to stop and start over if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas. Fundamental Identities: 1. Reciprocal Identities sin x = ( 1 / csc x ) csc x = ( 1 / sin x ) cos x = ( 1 / sec x ) sec x = ( 1 / cos x ) tan x = ( 1 / cot x ) cot x = ( 1 / tan x ) 2. Quotient Identities tan x = ( sin x / cos x ) cot x = ( cos x / sin x ) 3. Pythagorean Identities sin 2 x + cos 2 x = 1 or sin 2 x = 1 cos 2 x or cos 2 x = 1 sin 2 x 1+ tan 2 x = sec 2 x or 1 = sec 2 x tan 2 x or tan 2 x = sec 2 x 1 1+ cot 2 x = csc 2 x or 1 = csc 2 x cot 2 x or cot 2 x = csc 2 x 1 4. Even-Odd Identities ( Negative Formulas) cos ( -x ) = cos x sin ( -x ) = - sin x tan ( -x ) = - tan x sec ( -x ) = sec x csc ( -x ) = - csc x cot ( -x ) = - cot x θ 5. Cofunction Identities ( If is in radians, replace 90 o with /2. ) sin = cos ( 90 o ) csc = sec ( 90 o ) cos = sin ( 90 o ) sec = csc ( 90 o ) tan θ = cot ( 90 o θ ) cot θ = tan ( 90 o θ ) 6. Periodic Properties of Sine and Cosine Functions θ π θ π Sin ( + 2 n ) Cos ( + 2 n ) 7. Periodic Properties of Tangent and Cotangent Functions θ π θ π Tan ( + n ) Cot ( + n ) π

7 PRE-CALCULUS SECTION 6-2 IDENTITIES Sum and Difference Formulas for Cosines and Sines β β β β β β β β β β β β cos( + ) = cos cos - sin sin cos( - ) = cos cos + sin sin sin ( + ) = sin cos + cos sin sin ( - ) = sin cos - cos sin Sum and Difference Formulas for Tangents β β β tan ( + ) = ( tan + tan ) / ( 1 - tan tan ) β β β tan ( - ) = ( tan - tan ) / ( 1 + tan tan ) PRE-CALCULUS SECTION 6-3 IDENTITIES Double-Angle Formulas θ θ θ θ θ θ sin (2 ) = 2 sin cos cos (2 ) = cos 2 - sin 2 or cos (2 ) = 2 cos 2-1 or cos (2 ) = 1-2 sin 2 θ θ θ tan (2 ) = ( 2 tan ) / (1 - tan 2 ) Half-Angle Formulas where the sign is determined by the quadrant in which ( sin ( /2) = ± ( (1 - cos ) / 2 ) cos ( /2) = ± ( (1 + cos ) / 2 ) tan ( /2) = ± ( (1 - cos ) / (1 + cos ) ) or tan ( /2) = ( 1 - cos ) / ( sin ) or Power Reducing Formulas /2) lies. tan ( /2) = ( sin ) / ( 1 + cos ) sin 2 = 1 - cos2 cos 2 = 1 + cos2 tan 2 = 1 - cos cos2 θ θ θ

8 PRE-CALCULUS SECTION 7-1 IDENTITIES Law of Sines: a = b = c or Sin A Sin B Sin C Sin A Sin B Sin C a = b = c Formula to find the Altitude (height) of a Triangle: h = b sin A or h = a sin B Steps for solving Oblique Triangles using the Law of Sines If you are given SAA or ASA 1. Find the unknown angle using the sum of the two known angles. 2. Apply the Law of Sines to find the unknown side lengths If you are given SSA (the Ambiguous Case ) 1. Using the given angle, the side length opposite this given angle and the other given side, apply the Law of Sines to find the unknown angle. (If this angle measure is not possible in a triangle then there is No Solution and the triangle cannot exist.) 2. Find the third angle by using the given angle the angle solved for in step Find the remaining side length using the Law of Sines. 4. To check to see if there is a second triangle that could be formed using the originally given information. Find the second possibility for the angle found in step 1 using the angle solved for in step Find the second possibility for the angle solved for in step two using the originally given angle and the angle found in step 4. If this angle is negative then there is not a second triangle which can be formed using the original information. If this angle is positive there is a second triangle which can be formed using the original information so proceed to step Apply the Law of Sines using the originally given angle measure, the originally give side length opposite the given angle and the angle solved for in step 5 to solve for the still unknown side length. Area of Oblique Triangles: Area = (1 / 2) b c sin A = (1 / 2) a b sin C = (1 / 2) a c sin B

9 PRE-CALCULUS SECTION 7-2 IDENTITIES Law of Cosines: a 2 = b 2 + c 2-2 b c Cos A b 2 = a 2 + c 2-2 a c Cos B c 2 = a 2 + b 2-2 a b Cos C Solving SAS Triangle: 1. Use the Law of Cosines to find the side opposite the given angle. 2. Use the Law of Sines to find the angle opposite the shorter of the two given sides. This angle is always acute. 3. Find the third angle by subtracting the measure of the given angle and the angle found in step 2 from Solving SSS Triangle: 1. Use the Law of Cosines to find the angle opposite the longest side. 2. Use the Law of Sines to find the angle opposite the shorter of the two given sides. (This angle is always acute.) 3. Find the third angle by subtracting the measure of the given angle and the angle found in step 2 from Heron s Formula for the Area of a Triangle: Area = s ( s - a ) ( s - b ) ( s - c ) where s = (1 / 2) ( a + b + c ) PRE-CALCULUS SECTION 7-3 IDENTITIES Relationships between Polar and Rectangular Coordinates: x = r cos θ y = r sin θ x 2 + y 2 = r 2 or r 2 = x 2 + y 2 r = x + y tan θ = ( y / x )