SOLVING TRIGONOMETRIC EQUATIONS Note: If you would like a review of trigonometry, click on trigonometry. Example 1: Solve for x in the following equation. There are an infinite number of solutions to this problem. First isolate the cosine term. To solve for x, we have to isolate x. Let's restrict the domain so the function is one-to-one on the restricted domain while preserving the original range. If we restrict the domain of the cosine function to that interval , we can take the arccosine of both sides of each equation. The angle x is the reference angle. Therefore, if , then The period of equals and the period of , this means other solutions exists every units. where n is an integer. The approximate values of these solutions are You can check each solution algebraically by substituting each solution in the original equation. You can also check the solutions graphically by graphing the function formed by subtracting the right side of the original equation from the left side of the original equation. Algebraic Check: Left Side:
Sine, Cosine and Tangent in Four Quadrants Sine, Cosine and Tangent The three main functions in trigonometry are Sine, Cosine and Tangent. They are easy to calculate: Divide the length of one side of a right angled triangle by another side ... but we must know which sides! For an angle θ, the functions are calculated this way: Example: What is the sine of 35°? Cartesian Coordinates Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is: The point (12,5) is 12 units along, and 5 units up. Four Quadrants When we include negative values, the x and y axes divide the space up into 4 pieces: Quadrants I, II, III and IV (They are numbered in a counter-clockwise direction) In Quadrant I both x and y are positive, in Quadrant II x is negative (y is still positive), in Quadrant III both x and y are negative, and in Quadrant IV x is positive again, and y is negative. Like this: Example: The point "C" (−2,−1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction). There is a pattern!
Graphing Trigonometric Functions Graphing Trigonometric Functions (page 1 of 3) Sections: Introduction, Examples with amplitude and vertical shift, Example with phase shift You've already learned the basic trig graphs. But just as you could make the basic quadratic x2, more complicated, such as –(x + 5)2 – 3, so also trig graphs can be made more complicated. Let's start with the basic sine function, f(t) = sin(t). Now let's look at g(t) = 3sin(t): Do you see that the graph is three times as tall? Now let's look at h(t) = sin(2t): Copyright © Elizabeth Stapel 2010 All Rights Reserved Do you see that the graph is squished in from the sides? For sines and cosines (and their reciprocals), the "regular" period is 2π, so the formula is: For tangents and cotangents, the "regular" period is π, so the formula is: In the sine wave graphed above, the value of B was 2. (Note: Different books use different letters to stand for the period formula. Now let's looks at j(t) = sin(t – π/3): Now let's look at k(t) = sin(t) + 3:
Algebra/Trig Review - Solving Trig Equations Solve the following trig equations. For those without intervals listed find ALL possible solutions. For those with intervals listed find only the solutions that fall in those intervals. There’s not much to do with this one. Just divide both sides by 2 and then go to the unit circle. So, we are looking for all the values of t for which cosine will have the value of . From quick inspection we can see that is a solution. . To find this angle for this problem all we need to do is use a little geometry. with the positive x-axis, then so must the angle in the fourth quadrant. , but again, it’s more common to use positive angles so, we’ll use We aren’t done with this problem. that we want for the solution and sometimes we will want both (or neither) of the listed angles. This is very easy to do. to represent all the possible angles that can end at the same location on the unit circle, i.e. angles that end at . So, all together the complete solution to this problem is by using in the second solution.
2.Symmetry We tend to think about symmetry in terms of geometry more than anything else. That's understandable; it's easy to fold shapes in half in different ways, to see if they match up. Well, functions can have symmetry too, and trig functions are like the sumo symmetry champs. There are two types of symmetry when we look at trig functions. It has symmetry over the y-axis. We don't have to look at a graph to show that a function is even. f(-x) = f(x) Take off the blindfold and take another look at the graph. We can check that cosine fits the equation by looking at the unit circle. A(ll) S(ine) T(angent) C(osine). The absolute value of cos a and cos (-a) will be the same, because they have the same reference angle. This Looks Odd To Us If there exists something called "even" functions, is it really any surprise that there are odd functions as well? Turns out that y = sin x is an odd function. f(-x) = -f(x) For sine (and its reciprocal, cosecant), we have this breakdown of signs: Oh, By the Way…
Amplitude, Period, Phase Shift and Frequency Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions. The Period is the length from one peak to the next (or from any point to the next matching point): The Amplitude is the height from the center line to the peak (or to the trough). Or we can measure the height from highest to lowest points and divide that by 2. The Phase Shift is how far the function is horizontally to the right of the usual position. The Vertical Shift is how far the function is vertically from the usual position. All Together Now! We can have all of them in one equation: y = A sin(Bx + C) + D amplitude is A period is 2π/B phase shift is −C/B vertical shift is D Example: sin(x) This is the basic unchanged sine formula. So amplitude is 1, period is 2π, there is no phase shift or vertical shift: Example: 2 sin(4x − 2) + 3 amplitude A = 2 period 2π/B = 2π/4 = π/2 phase shift −C/B = −(−2)/4 = 1/2 vertical shift D = 3 In words: Note the Phase Shift formula −C/B has a minus sign: And we get: Frequency
Higher Bitesize Maths - Radians and equations : Revision, Page4 Chapter 5: Trigonometric Functions<BLURT> Chapter 5: Trigonometric Functions 1. Please solve 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Copyright ©2001 The McGraw-Hill Companies. Algebra - Symmetry Example 1 Determine the symmetry of each of the following equations. (a) [Solution] (b) (c) (d) (e) Solution (a) We’ll first check for symmetry about the x-axis. Now, this is not an equivalent equation since the terms on the right are identical to the original equation and the term on the left is the opposite sign. Next, let’s check symmetry about the y-axis. After simplifying we got exactly the same equation back out which means that the two are equivalent. Finally, we need to check for symmetry about the origin. So, as with the first test, the left side is different from the original equation and the right side is identical to the original equation. [Return to Problems] (b) We’ll not put in quite as much detail here. We don’t have symmetry here since the one side is identical to the original equation and the other isn’t. Next, check for symmetry about the y-axis. Remember that if we take a negative to an odd power the minus sign can come out in front. (c) Now, check for symmetry about the y-axis.
Symmetry and Graphing Symmetry and Graphing (page 3 of 3) Sections: Symmetry about an axis, Symmetry about a point, Symmetry and graphing Symmetry is more of a geometrical than an algebraic concept, but the subject of symmetry does come up in a couple of algebraic contexts. For instance, when you're graphing quadratics, you may be asked for the parabola's axis of symmetry. This is usually just the vertical line x = h, where "h" is the x-coordinate of the vertex, (h, k). The other customary context for symmetry is judging from a graph whether a function is even or odd. For each of the following graphs, list any symmetries, and state whether the graph shows a function. Graph A: This graph is symmetric about its axis, the line x = 3. Graph B: This graph is symmetric about the axes x = 0 (the y-axis) and y = 0 (the x-axis), and also about the origin. Graph C: This graph is symmetric about the axes x = 1 and y = –2, and symmetric about the point (1, –2). Graph G: This parabola is lying on its side.