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Transcription

1 Unit Analytical Trigonometry Classwork A) Verifying Trig Identities: Definitions to know: Equality: a statement that is always true. example:, + 7, 6 6, ( + ) Equation: a statement that is conditionally true, depending on the value of a variable. example: x +, ( x ), x x + x 0, sin. Identity: a statement that is always true no matter the value of the variable. example: x + x x, ( x ) x, ( x ) x x +, x x + x. In the last example, it could be argued that this is x not an identity, because it is not true for all values of the variable (x cannot be or -). However, when such statements are written, we assume the domain is taken into consideration although we don t always write it. So a better definition of an identity is: a statement that is always true for all values of the variable within its domain. The 8 Fundamental Trigonometric Identities: Trig Identities proofs (assuming in standard position) Reciprocal Identities csc sin Quotient Identities tan sin cos Pythagorean Identities sec cos cot cos sin cot tan sin r y y csc r cos r x x sec r tan x y y cot x y sin cos r y x tan x r x cos sin r x y cot x + y r x + y r x + y r sin + cos x + tan sec r + y r r x r x + y x r x x y + y y r y + cot csc cos + sin + tan sec cot + csc Corollaries: a statement that is true because another statement is true: Examples (you write the others): y r Reciprocal identities: sin csc sin csc sin cos Quotient identities: tan cos sin cos sin tan Pythagorean identities: sin cos cos sin sin ± cos cos ± sin. Analytical Trigonometry Stu Schwartz

2 In this section, you will be given a number of trigonometric identities. Remember they are true. Your job will be proving that they are true. Your tools will be your knowledge of algebra, the 8 trig identities, and your ingenuity. Some are easy like example and others are more difficult like example. Example ) sin ( csc sin ) cos Example ) sec x sin csc sin sin cos sec x csc tan x sec x sec x cosx sec x sec x sec x csc x tan x Guidelines for verifying trigonometric identities: ) Your job is to prove one side of an identity is equal to the other so you will only work on one side of the identity, so ) Always work on the most complicated side and try to transform it to the simpler side. More complicated can mean the side that is longer or has more complicated expressions. Additions (or subtractions) are generally more complicated than multiplications. ) If an expression can be multiplied out, do so. ) If an expression can be factored, do so. ) If you have a polynomial over a single term, you can split it into several fractions. 6) If you have an expression, that involves adding fractions, do so finding a lowest common denominator. 7) When in doubt, convert everything to sines and cosines. 8) Don t be afraid to create complex fractions. Once you do that, many problems are a step away from solution. 9) Always try something! You don t have to see the solution before you actually do the problem. Sometimes when you try something, the solution just evolves. ) csc x + sec x sec x ) cos x + sin x cos x + csc x + sin x sec x + cos x + tan x sec x cos x + cos x + sin x cos x + ) cos x sin x 6) + sin x sin x sin x cosx + ( ) sin x + + cos x + sin x + cos x sin x + cos x + sin x + cos x 7) cot x csc x cosx 8) tan x + cot x sec x csc x + sin x + cos x sec x csc x. Analytical Trigonometry Stu Schwartz

3 9) sec x tan x 0) + cot x csc x cos x ) ) sin x tan x cot x + + tan x cot x tan x + tan x tan x ( tan x ( tan x + tan x tan x + + sec x ) + + ( + ) sin x cos x sec x + sin x + cos x csc x ) sec x sin x sec x sec x cos x sin x csc x + cot x cot x csc x tan x + + cosx + cosx + cos x sin x + sin x B) Sum and difference Formulas Determine whether the sine function is distributive: that is sin A + B values of A and B. Check out whether sin ( cot x csc x + sin x + sin A + sin B. Let s try it with different sin0 + sin60. + There are geometric proofs to determine the sum and difference formulas for trig functions: sin( A + B) sin AcosB + cos Asin B cos( A + B) cos AcosB sin Asin B tan( A + B) sin( A B) sin AcosB cos Asin B cos( A B) cos AcosB + sin Asin B tan A B tan A + tan B tan Atan B tan A tan B + tan Atan B Example ) Find the exact value of sin7 sin 0 + sin0 cos + cos0 sin Example ) Find the exact value of tan7 in two ways. tan7 sin tan +tan 0 cos7 + 6 tan tan 0 Example ) Find the exact value of cos7 cos 0 + cos0 cos sin0 sin ( ( 6 Example ) Find the exact value of tan tan tan tan 0 +tan tan 0 +. Analytical Trigonometry Stu Schwartz

4 Example ) Given sin A a. sin A + B and cosb, both A and B in quadrant I, find b. cos( A + B) c. tan( A + B) d. quadrant of ( A + B) ( ( 6 6 Example 6) Given cos A 7, A in quadrant IV and cosb a. sin( A B) b. cos A B,B in quadrant II, find quadrant II c. tan( A B) d. quadrant of ( A B) 7 ( ( 7 ( + ( ( quadrant II Example 7) Verify that sin( x + 90 ) Example 8) Verify that tan( x +80 ) tan x sincos90 sin90 C) Double Angle formulas Recall that sin A + B tan x tan80 tan x80 tan x sin AcosB + cos Asin B. If A B, we get sin( A + A) sin Acos A + cos Asin A So sina sin Acos A. This works for the other trig functions as well getting the double angle formulas. sina sin Acos A cosa cos A sin A or cos A or sin A tana tan A tan A Example ) Using trig functions of 0, find the values of: a) sin60 b) cos60 c) tan60 sin 0 cos 0 cos 0 sin 0 tan 0 tan 0 Example ) Given sin A, A in quadrant I find a. sina b. cos A c. tana d. quadrant of A sin Acos A cos A sin A quadrant II. Analytical Trigonometry Stu Schwartz

5 Example ) Given tan A, A in quadrant II find a. sina b. cos A c. tana d. quadrant of A sin Acos A cos A sin A ( ( 9 quadrant IV Example ) Express sinx in terms of the angle x. Example ) Verify the following identities: sinx cosx ( ) ( cos x sin x) cos x sin x a) sinx sec x sec x b) ( ) sinx sin x + cos x sinx D) Half-angle formulas: These formulas are more obscure and are not used that much. Still, you should know that they exist and be able to use them. sin A ± cos A cos A ± + cos A tan A cos A sin A or sin A + cos A The signs of sin A and cos A depend on the quadrant in which A lies. Example ) Find the exact values of the following using half-angle formulas. a) sin b. cos c) tan cos0 + cos0 + Example ) Given sin A, A in quadrant III find + + a. sin A b. cos A c. tan A d. quadrant of A quadrant II. Analytical Trigonometry Stu Schwartz

6 E) Solving trigonometric equations Just as we solved equations for a value of x that satisfied the equation, we do the same for trig equations in this case finding the value of an angle that satisfies the equation. Example ). We can do this by inspection we know from our knowledge of graphing and quadrant angles that the angle that satisfies this equation is 90 or (. However, there are other angles that satisfy this equation like 0,80,70,... So we usually solve the equation on a certain domain. Usually we will solve it on 0 x < 60 or 0 x <. We can also verify our solutions by graphing the equation. Example ) + 0 x 0,0 Example ) sin x sin x 0 x 0,80,90 Example ) Example ) sin x tan x x, sin x ± x,,, Example 6) sin x + 0 Example 7) cos x ( )( ) 0 No solution x 0,0 cos x 0 ( +) 0 x 0,0,0 Example 8) Example 9) sinx 0 ( ) cos x sin x + cos x sin x + sin x sin x 0 0 x 0 /,80, x 0,80,90,70. Analytical Trigonometry Stu Schwartz

7 Unit Analytical Trigonometry Homework. Verify the following identities: There are additional problems in your book. a) csc x( cos x) b) ( + ) ( ) sin x + + cos x ( sin x + cos x) sin x + + c) ( csc x + sec x) sec x d) cot x + csc x + + cos x + tan x sec x csc x + csc x + e) cos x cos x sin x sin x f) tan x + sec x cos x( cos x ) ( sin x) (sin x) sin x sin x g) csc x + sin x + cos x sec x h) sec x sec x sin x cos x i) + cot x csc x j) + sin x + cos x csc x sec x ( ) sec x sin x + cos x sec x. Analytical Trigonometry Stu Schwartz

8 k) + + cos csc x l) + tan x m) ( ) cos x sin x csc x ( ) ( ) + + sin x cos tan x + + csc x + n) sin x cos x sin x o) tan x + ( ( ) sin x tan x p) tan x tan x ( ( ) tan x sin x sec x + r) cos x q) + tan x + tan x + tan x sec x + tan x ( sin x cos x) sin x + cos x [ sin x ( sin x) ] sin x + sin x sin x tan x sec x + sec x tan x tan x sec x + ( ( sec x sec x + sec x tan x sec x + tan x tan x sec x + sec x + tan x sec x tan x + cot x cosx + cosx cosx sin x + cos x cos x + sec x + + s) cos x sin x cos x t) ( cos x sin x) ( cos x + sin x) + ( ( cos x ( cos + x) + cos x cos x + ( + ) sin x. Analytical Trigonometry Stu Schwartz

9 . Find the exact values of the following expressions. Make appropriate pictures. a. sin0,cos0,tan0 b. sin,cos, tan 0 ( 60 + ) By formulas : sin0 6 + cos0 6 tan ( + 0 ) - there are other combinations By formulas : sin 6 cos 6 + tan Given sin A a. sin A + B 8 and cosb, both A and B in quadrant I, find 7 b. cos( A + B) c. tan( A + B) d. quadrant of ( A + B) sin AcosB + cos AsinB cos AcosB sin AsinB ( ( ( ( Given cos A, A in quadrant III and cosb,b in quadrant I, find a. sin A + B Quadrant b. cos( A + B) c. tan( A + B) d. quadrant of ( A + B) sin AcosB + cos AsinB ( ( + ( 0 ( cos AcosB sin AsinB ( ( ( + 0 ( +. Given tan A, A in quadrant III and sinb,b in quadrant II, find Quadrant a. sin( A B) b. cos( A B) c. tan( A B) d. quadrant of ( A B) sin AcosB cos AsinB ( ( 6 6 ( ( cos AcosB + sin AsinB ( 6 ( + 6 ( ( + 0 Quadrant Analytical Trigonometry Stu Schwartz

10 6. Verify the following identities: a) cos( 70 x) b) sin( x 0 ) ( ) cos0 sin0 cos70 + sin70 0 ( ( ) c) sin( x ) + tan x + ) 0 d) tan( x + 60 ) ( tan x tan x cos sin cos sin tan x + tan60 ) tan x tan60 ( tan x + tan x ( ( tan x tan x tan x e) sin( A + B)sin( A B) sin A sin B f) cos( A + B) + cos A B cos AcosB ( sin AcosB + cos AsinB) ( sin AcosB cos AsinB) sin Acos B cos Asin B sin A( sin B) ( sin A)sin B sin A sin Asin B sin B + sin Asin B sin A sin B cos AcosB sin AsinB + cos AcosB + sin AsinB cos AcosB 7. Using trig functions of 60, find the values of a) sin0 b) cos0 c) tan0 sin0 sin( 60 ) cos0 cos( 60 ) tan0 sin0 sin60 cos60 cos 60 sin 60 cos0 ( ( ) ) ( ( 8. Given sin A 7, A in quadrant II find a. sina b. cosa c. tana d. quadrant of A sin Acos A cos A sin A sina cosa 7 ( (6 ( 7 ( 7 sina 6 6 cosa 6 sina, cosa + Quadrant IV 7 9. Given cos A, A in quadrant I find 0 a. sina b. cosa c. tana d. quadrant of A sin Acos A cos A sin A sina cosa sina +, cosa ( ( 9 sina cosa 99 Quadrant II 9. Analytical Trigonometry Stu Schwartz

11 0. Given tan A, A in quadrant III find a. sina b. cosa c. tana d. quadrant of A sin Acos A cos A sin A sina cosa ( ( ( ( sina cosa sina +, cosa Quadrant II. Verify the following identities: a. cos x sin x cosx b. cscx sec x csc x ( cos x sin x) ( cos x + sin x) ( cos x sin x) cosx c. sinx sin x d. cosx cos x + x cosx cos x sinx + cosx + ( sin x) cos x + sin x + sin x sin x sin x + sin x sinx csc x sec x cosx sinx ( cos x ) ( ) cos x sin x cos x ( cos x) cos x + cos x sin x cos x. Find the exact values of the following using half-angle formulas. a) sin. b. cos. c) tan. cos. Given sin A 8, A in quadrant II find 7 + cos a. sin A b. cos A c. tan A d. quadrant of A cos A cos A Given cos A, A in quadrant IV find 6 Quadrant I a. sin A b. cos A c. tan A d. quadrant of A cos A 6 + cos A Quadrant. Analytical Trigonometry Stu Schwartz

12 . Solve the following equations on [ 0,60 ) a) b) cos x 0 x 60,00 ( ) 0 0 x 90,70 x 0 c) sin x d) cos x sin x ± x 60,0,0,00 cos x ± x,,, e) cos x f) tan x ( ) 0 0 x 90,70 x 60,00 tan x tanx x, x, f) cos x h) sin x ( +) ( + ) 0 x 0,0 No solution ( )( +) 0 0 x 0,80,90,70 i) cos x sin x j) sin x sin x ± x 0,0,0,0 cos x + sin x sin x + sin x 0 x 0,/ 8 / 0 / /,90 k) sinx l) cosx ( ) 0 0 x 90,70,0,0 cos x ( +) 0 x 0,0,0. Analytical Trigonometry Stu Schwartz