Friday, May 29, 2009

Trig Identities & Solving Trig Equations

Trig Functions in Terms of Each Other & Identities


The Pythagorean Identity:


All of these equations can be used to simplify an equation or to make sure there is only one trigonometric expression, that way they're so much easier to work with!

And now for some examples!

If cos(x) is .25, then sec(x) is 4.

If cos(x) is 1/5 and sin(x) is 2/5, then cot(x) is 1/2.

Verifying Trig Equations

And now it's time to learn how to verify trig identities! - a.k.a. TRIG PROOFS! (insert ominous music here)

To do this you are given two things that are already equal to each other but are still forced to prove that they are equal through the use of tiny baby steps. The end goal of any trig identity verification is to make the left side of the equation and the right side of the equation look the same.

HOWEVER, this must be done with extreme caution, for ONLY ONE SIDE OF THE EQUATION CAN BE CHANGED. If you ever even think of messing around with the other side of the equation, or even going so far as to CROSS-MULTIPLY, you will be promptly cast into the fearsome pit of trigonometric DOOM!!!!

In order to become a trig proof master, one must follow THE RULES (Actually, they're more like guidelines):

1. Don't lose track of what you are trying to prove. Randomly swapping in trig expressions for other trig expressions doesn't usually work. A better plan would be to look at what the end result should be before diving in. Verifying trig identities is not guess and check. (For an example of guess and check, see 'finding the roots of a polynomial.')

2. Work on making the more complicated side less complicated (or rewrite everything in terms of sine and cosine). Making something more complicated than it already is is generally a waste of time.

3. Work from minor functions (csc, sec, cot, tan) to major functions (sin and cos). This makes sense when you realize that sin and cos are a lot less complicated. (See rule #2)

4. Use the common denominator. When you find a common denominator, you can put two fractions together to form one fraction. And, believe it or not, one fraction is in fact less complicated than two fractions (See rule #2). I know, it's weird, but 1 really is less than 2.

5. Use conjugates. Remember how, in the first section of this post, sin θ was equal to 1 / csc θ? Guess what? It still is! And guess what else? In general, sin θ is a lot less complicated than 1 / csc θ. (Rule #2) Is this starting to sound familiar?

6. UseDid you notice that the left hand side is one fraction and the right hand side is two? And did you notice that they all have a common denominator? Does anyone remember rule #4?

7. Use the Force... just checking to see if anyone was paying any attention.

8. Try not to take the square roots of things or raise them to powers. Remember how we're only supposed to change one side of the equation? If you suddenly make one side exponentially bigger or smaller, what are you supposed to do with the other side to make them match up?

9. Remember:

Last time I checked, trying to put two fractions together by finding a common numerator was not considered to be a very good idea. People who do this are getting way too overexcited about rule #4.

10. A trig proof is like an Oreo cookie. There is always more than one way to verify a trig identity. If your first attempt fails miserably, don't be afraid to go back and try something new!


Now let's try a sample identity:

Reminder: Sometimes it is ACCEPTABLE to alter both sides in order to verify a trig proof! However, this should only be done if it SAVES MUCH TIME. Never add or remove anything from either side, or you will suffer chastisment of GREAT VEHEMENCE.

Equation Solving

Equations get MUCH harder once you actually have to figure out what theta or x is.
For example:

1. The goal is to get the trigonometric expression by itself. So in this case:

2. Next grab your calculator and take the arc of that expression by pressing [2nd], and then the trigonometric expression in the equation:

3. REMEMBER!!! This is only the reference angle and sin(x) & cos(x) are positive in two quadrants when the period is "0" is less than "x" is less than "360". Keep in mind that tan(x) only has a period of "0" is less than "x" is less than "180".
  • sin(x) pos. in Quadrants I,II

  • cos(x) pos. in Quadrants I, IV

  • tan(x) pos. in Quadrants I, III
The final answer for this problem would be 45 & 315 degrees.

  • If you come across something like sin(2x) then "0" is less than "2x" is less than "360", so "0" is less than "x" is less than "180".

  • Sometimes the question will ask for a general solution. This is in the form of {ref. angle + n ● period} where 'n' is an integer.
  • Trickier problems involve factoring and replacing some trigonometric expressions with others so only one ends up in the equation
  • Radians are also commonly used instead of degrees. Just know that pi = 180 degrees

Equation Solving on T.I.

To solve a trig identity on a TI:

  1. Equip your TI Graphing calculator.
  2. Hit mode and set to radians.
  3. Input the function into Y=.

  4. Use zoom fit to get appropriate window, or set it manually.
  5. Hit [Calc], then select "zero", select a point with a positive y value, then a point with a negative y value, then hit enter to find the roots of the function for one period. Avoid ERROR: BAD GUESS.

**Round your answers to the nearest ten thousandth.
**Don't think you're not supposed to write down the viewing window you used.

Sum and Difference Formulas

**With ( A - B ) just reverse the addition/subtraction signs.

Double Formulas

Practice Questions
1) Verify:

2) Verify:

3) Solve for 1 period:

4) Solve for all solutions/general solution (nЄZ):

5) Determine if Schrodinger's cat is alive or dead

6: Derive sin(3x).

7: Verify

8: Solve for 1 period:

9: Solve for (nЄZ):

10: Solve for x using your TI graphing calcy, 0<= x <= 2pi:

11: If sin(x) = 4/5, sec(x) = ?

12: Verify:





5) Kitteh was dead before he was even placed in box!





10: x = {.793, 1.385, 4.530, 5.634}

11: 5/3


Trig Functions and Graphs

A radian is a way to measure angles based on how many radiuses lie along the arc intercepted by the central angle. For an angle in radians, where S = arc length:

Since there are pi radians in half a circle, and also 180 degrees in half a circle, you can convert degrees to radians and vice versa with the following proportion:

Also, we are expected to be able to work with revolutions. Recall that 1 revolution = 1 circumference = 360 degrees = 2pi radians.

Since rates are determined by the amount of change divided by time, the following formulas apply:

The Unit Circle

For any point on the unit circle, the x coordinate is the cosine of the central angle, and the y coordinate is the sine of the central angle. The following values should be memorized. Each point (x, y) indicates (cos, sin) of the central angle. (diagram source: The PreCal40S Blog)

Graphs of Trig Functions

Sine and Cosine graphs come in this format: y = a sin (bx) + c where a is the amplitude, b is the frequency (how many complete cycles in 2pi), and c is the vertical shift. The period, or the length of one cycle, can be obtained by .

For example, here is the graph of y = -3 cos (2x) + 4 from -2pi to 2pi, which has a period of pi:

Practice Problems

1. Write in degrees and in number of revolutions.

2. In radians, find the central angle of a 9 cm arc in a 6 cm circle.

3. The second hand on a clock is 6 inches long. Find the speed of the tip of the second hand.

4. Give the period of each trig function: y = sin x, y = cos x, y = tan x.

5. Describe each trig function as odd or even: y = sin x, y = cos x, y = tan x.

6. Find in radians in terms of pi.

7. Find in radical form.

8. Write an equation for a sinusoidal graph with a minima at 0, a maxima at 4, and a period of pi.

9. What is the range of the function y = -5cosx - 3 ?

10. Find the linear speed in cm/sec and angular speed in radians/sec of a point on the edge of a rotating compact disk that makes 500 revolutions per minute. A standard cd is 12 cm in diameter.

(Solutions Pending)