Trigonometry - August 20, 2010 1
Proble Problem m So Solvi lving ng Strat Strategi egies es
• Graphs can help provide intuition for problem involving trig functions. • Some problems involving inverse trig functions can be most easily solved by rewriting the problem in terms terms of the corresponding corresponding trig functions. functions.
• When stuck on a trig problem involving tangent, cotangent, secant, and or cosecant, try rewriting the problem in terms of sine and cosine.
• Solving a simpler version of a complicated problem can help solve the original complex problem. • In equations involving trig functions, try to isolate trig functions by manipulating the equation so that you have a trig function equal to a constant.
• Trig equations with a single type of trig function are often easier to deal with than equations with two different types of trig functions.
• If a tactic makes some progress on a problem, but doesn’t entirely solve it, look for a way to use the tactic again.
• Don’t get locked into one way of attacking a problem-look for several starting points, go a little ways down each path, and then choose the one that looks most fruitful.
• Wishful thinking is a powerful problem solving tool-thinking about what you’d like to be true can focus your attention on the key step in a solution.
• A fruitful first step in many trig problems is rewriting rewriting the problem in terms terms of sine and cosine, because most people have more experience with sine and cosine than other trig functions.
• When solving an equation that contains trig functions, it’s often best to write the equation in terms of a single trig function, so that you can then try to solve the equation for that function.
• Recognizing the forms of common trigonometric identities can help solve many trigonometric problems. • Recognizing Recognizing relationships relationships among angles in a complicate complicated d trig expression expression can help simplify simplify the expression.
• Many series involving trigonometric functions can be written as telescoping series. • Trigonometry can be a powerful tool even in problems that don’t appear to be about trigonometry.
1
2
Trigonometric Identities
sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ cot2 θ + 1 = csc2 θ sin α
± β = sin α cos β ± sin β cos α cos α ± β = cos α cos β sin α sin β tan α ± tan β tan α ± β = 1 tan α tan β sin2x = 2 sin x cos x cos2x = cos2 x tan2x =
2
− sin
x = 2 cos2 x
2
− 1 = 1 − 2sin
x
tan2x 1 tan2 x
− 1 + cos θ θ cos = ± 2 2 θ 1 − cos θ sin
= 2 2 θ sin θ 1 cos θ tan = = 2 1 + cos θ sin θ
±
−
2
1. What is the period of f (x) = (sin x)(cos x)? 2. How many values of x satisfy sin x =
2 x
625
?
3. For what values of θ is sin 3θ < 0.5? 4. While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. What are the two smallest positive values of x such that the sine of x degrees equals the sine of x radians? 5. The price of Up-N-Down Stock varies sinusoidally with time. At the start of this year, the price of the stock is $110 per share. Over the next 3 days, the price climbs to a peak of $120 per share. If the lowest price the stock reaches is $80 per share, how much money can you make this year (in 365 days starting from the start of this year) if you start with $1000 and time all of your trades perfectly? (Assume that you can buy fractions of a share of stock, and that you can buy stock as many times as you want, but can never sell stock you haven’t already bought first). 6. Find the smallest positive integer solution to
cos96 + sin 96 . cos96 sin96 ◦
tan19x = ◦
◦
◦
−
◦
7. Determine A + B if A and B are acute angles such that
sin A + sin B =
8. Find the sum of the roots of tan 2 x
3
and
2
cos A
− cos B =
1 2
− 9tan x + 1 = 0 that are between x = 0 and x = 2π radians.
9. Let a = π/2008. Find the smallest positive integer n such that
2[cos a sin a + cos 4a sin2a + cos 9a sin3a + ... + cos (n2 a)sin(na)]
is an integer. 10. Let a,b,c,d be positive real numbers such that
a2 + b2 = 1 c2 + d2 = 1 1 ac bd = . 2
−
3
Calculate ad + bc. 11. Evaluate
cos1 + cos 2 + cos 3 + ... + cos 43 + cos 44 . sin1 + sin 2 + sin 3 + ... + sin43 + sin 44 ◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
12. Given any seven real numbers, prove that there are two of them, x and y, such that
0
13. Evaluate (sin1 )(sin 3 )(sin 5 ) ◦
◦
◦
≤ 1x+−xyy ≤ √13 .
· ·· (sin 177 )(sin 179 ). ◦
◦
14. Find the value of tan x if sin x + cos x = 1/5 and π/2 < x < π.
15. If f (
x x−1
)=
1 x
for all x = 0, 1, and 0 < θ < π/2, then find f (sec2 θ).
16. Compute the smallest positive angle x, in degrees, such that tan 4x =
4