Chapter 7: Analytic Trigonometry

Transcription

1 Chapter 7: Analytic Trigonometry 7. Trigonometric Identities Below are the basic trig identities discussed in previous chapters. Reciprocal csc(x) sec(x) cot(x) sin(x) cos(x) tan(x) Quotient sin(x) cos(x) tan(x) cot(x) cos(x) sin(x) Pythagorean sin (x) + cos (x) + tan (x) sec (x) + cot (x) csc (x) Even-Odd sin( x) sin(x) cos( x) cos(x) tan( x) tan(x) Cofunction (the function of angle u equals the cofunction of the complementary angle) π π π sin u cos( u) tan u cot( u) sec u csc( u) π π π cos u sin ( u) cot u tan ( u) csc u sec( u) These may be used to establish proofs of other identities, to solve trig equations, or to simplify problems in calculus. In order to prove the identity below, we work to turn the LHS into the RHS. We agree to work on one side only of the identity. Often it is helpful to put everything in terms of sin and cos and find common denominators. + csc(x) Example : Pr ove sec(x) cos(x) + cot(x) sin(x) + + sin(x) sin(x) cos(x) cos(x) cos(x) sin(x) + cos(x) + sin(x) sin(x) sin(x) + sin(x) sec(x) sin(x) cos(x) sin(x) cos(x) [ + ]

2 cos(x) sin(x) Example: Prove. Work with the RHS and multiply by conjugate. sin(x) + cos(x) sin(x) cos(x) sin(x) [ cos(x) ] sin(x) [ cos(x) ] + cos(x) cos(x) cos (x) sin (x) cos(x) sin(x) Example: Substitute x sin(θ) into the following expression and simplify. Take 0 θ < π/ 9 x 9 sin( θ ) 9 9sin ( θ) 9 sin ( θ ) 9 cos ( θ ) cos( θ) Note: you must memorize the basic identities. You need not write the reasons for each step, but show every step. Remember you are only allowed to work on one side of an identity. You are not solving an equation; do not move quantities from one side to another, etc.

3 7. Addition and Subtraction Formulas ( + ) + ( ) ( + ) ( ) + ( tan ( x) + tan ( y) + y) tan ( x) tan ( y) ( tan ( x) tan ( y) y) + tan ( x) tan ( y) sin x y sin x cos y cos x sin y sin x y sin x cos y cos x sin y cos x y cos x cos y sin x sin y cos x y cos x cos y sin x sin y tan x tan x These are proved in the text or left for exercises. Example: Find exactly sin(5 ) sin(π/) sin 5 sin 45 0 sin 45 cos 0 cos 45 sin π π π π sin u sin cos( u) cos sin ( u) cos u 0sin u cos u Example: Prove sin u cos( u) Another helpful identity converts a sum of sin and cos into a single sin function. For A and B real numbers, let k A + B and let φ satisfy A B cos( φ ) and sin ( φ ) A + B A + B A sin x + B cos x k sin x +φ Then

4 7. Double Angle, Half Angle, and Product Sum Formulas Double Angle sin x sin x cos x ( tanx ) tan ( x) cos x cos x sin x sin x cos x tan x Lowering Powers cos( x) cos( x) cos x sin ( x) + cos ( x) tan ( x) + cos x Half Angle: Choose the correct sign for the quadrant of x/ x x cos x + cos x sin ± cos ± x cos x sin x tan sin x + cos x Product to Sum sin ( x) cos( y) sin ( x y) sin ( x y) + + cos( x) sin ( y) sin ( x y) sin ( x y) + cos( x) cos( y) cos( x y) cos( x y) + + sin ( x) sin ( y) cos( x y) cos( x + y) Sum to Product x+ y x y sin ( x) + sin ( y) sin cos x+ y x y sin ( x) sin ( y) cos sin x+ y x y cos( x) + cos( y) cos cos x+ y x y cos( x) cos( y) sin sin Proofs may be found in the text or are left as exercises for the reader. 4

5 Example: Prove tan ( x) + tan x tan x tan x tan x tan x + tan ( x) tan x tan ( x) tan ( x + x) tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x tan x 5

6 7.4 Inverse Trigonometric Functions You are given / sin(x) and asked to find x. You might try a graphical approach, looking for the intersection(s) of y / and y sin(x) There seem to be many possible solutions. This is similar to knowing /4 x and finding x ±/. Such situations are very confusing to calculators, which desperately want to return an answer but can only return a single value. We solved this problem by introducing a principal square root function ( ) as the inverse of y x that only returns the positive root. Essentially we restricted the domain of y x to only 0 x < so that the range of y x would be the same. Thus we made a one-to-one function of the restricted square function that had as an inverse the principal square root function. Let's try something similar. We can restrict the domain of y sin(x) so that the resulting function (called the restricted sine function y Sin(x) - note the capital S, it's quite sensitive about that) is one-to-one. There are many ways to do this. Do it my way, and we'll all get along famously. Choose the domain of y Sin(x) to be [ π/, π/]. Its range is [, ], and as you can see below, it is one-to-one. For this function, define an inverse function with domain [, ] and range [ π/, π/]. Recall the x and y values swap between function and inverse, their graphs being reflections in the line y x. What to call the inverse of the restricted sine function? Back in the day, when everyone was taking trig functions of angles, the notation was y Arcsin(x), meaning y was the angle whose Sine was x. The math went all loopy and decided to take trig functions of real numbers, and the (bad) notation that evolved has us today calling the inverse function y Sin (x). The in the exponent is not a real exponent - it just signifies inverse. In particular, Sin (x) / Sin(x). 6

7 y y ArcSin@xD

8 Now when you ask your calculator to find sin (/) it reports 0 or π/6 radians. It always returns an angle in I if your x value is positive, and an angle in IV if your x value is negative. (Try it: sin ( /) 0 or π/6 radians.) Note: Sin(Sin (x)) x, for x and Sin (Sin(x)) x, for π/ x π/ Be careful of the domains! Example: Sin (sin(0 ))? Not 0 - that's not in the domain of Sin. Proceed as follows: Sin ( /) 0. These inverse functions must always return a value from 90 to +90. What about inverses for all the other trig functions? It is all very similar to what we have done for Sin, but some of the domain restrictions may at first glance seem a bit odd. Let's just assume there is some sort of "useful in calculus" reason and make the best of it. y Cos (x) or y Arccos[x] has domain [, ] and range [0, π] Note this function return values in I (for +x) or II (for x) y Tan (x) or y Arctan(x) has domain [, ] and range ( π/, π/) y Csc (x) or y Arccsc(x) has domain (, ] [, ) and range (0, π/] [π, π/) y Sec (x) or y Arcsec(x) has domain (, ] y Cot (x) or y Arccot(x) has domain [, ] and range (0, π) [, ) and range [0, π/) (π, π/] Graphs of these functions may be found in the text. Note there are no keys on the TI-86 corresponding to the last three functions above. To find Sec (), think sec(θ) / cos(θ) Thus you wish to find the angle θ such that cos(θ) / or Cos (/) 60. Example: Find Cos ( /). This should return an angle in II and it does: 0 or π/ radians Example: Find cos(sin (/5)) exactly. We want cos(θ), where θ Sin (/5) or sin(θ) /5. 4 We could use an identity: cos( θ ) ± sin ( θ ) ± (choose + as Sin (+) in I) 5 5 However, a sketch of a right triangle with side opposite θ, hypotenuse 5, and hence adjacent to θ having side of 4 let's us see the cos(θ) 4/5 easily. 8

9 7.5 Trigonometric Equations Equations with trig functions are to be solved for the numbers or angles that make them true. Example: Solve cos(θ) 0. Isolate a single trig function as you would in solving a regular equation: cos(θ) / There are infinitely many solutions; between 0 and 60, both 60 and 00 have cos /. We write: θ n or n, where n is any integer. Alternatively, if solving for real numbers or angles in radians: π/ + πn or 5 π/ + πn. Example: Solve sin (x) 4 5 cos (x) for x on the interval [0, π). Recall sin (x) cos (x). The equation becomes: cos (x) 4 5 cos (x) 4 cos (x) cos(x) ± / x π/6, 5π/6, 7π/6, and π/6. Example: Solve sin( x) cos( x) Be careful when dividing both sides of an equation by a variable. Check that if the variable is 0, you are not losing solutions. Dividing both sides here by sin(x) is safe; no values that makes sin(x) 0 also make cos(x) 0, so we are losing no solutions. cot ( x) or tan ( x) x n or x n (Recall the periods of tan and cot are 80.) Thus x n or n This collapses to x n, where n is any integer. Example: Solve sin(x) x graphically. Graph both, and use ISECT on the TI We find x , 0, and satisfy the given equation. 9