 # Solving Trigonometric Equations Solving Trigonometric Equations (page 1 of 2) Solving trig equations use both the reference angles you've memorized and a lot of the algebra you've learned. Be prepared to need to think! Solve sin(x) + 2 = 3 for 0° < x < 360° Just as with linear equations, I'll first isolate the variable-containing term: sin(x) + 2 = 3 sin(x) = 1 Now I'll use the reference angles I've memorized: x = 90° Solve tan2(x) + 3 = 0 for 0° < x < 360° There's the temptation to quickly recall that the tangent of 60° involves the square root of 3 and slap down an answer, but this equation doesn't actually have a solution: tan2(x) = –3 How can the square of a trig function evaluate to a negative number? no solution Solve on 0° < x < 360° To solve this, I need to do some simple factoring: Now that I've done the algebra, I can do the trig. x = 30°, 90°, 270°, 330° Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved Solve sin2(x) – sin(x) = 2 on 0° < x < 360° Only one of the factor solutions is sensible. x = 270° Hmm...

Amplitude and Period Amplitude and Period Learning Objective(s) · Understand amplitude and period. · Graph the sine function with changes in amplitude and period. · Graph the cosine function with changes in amplitude and period. · Match a sine or cosine function to its graph and vice versa. You know how to graph the functions and . or , where a and b are constants. We used the variable previously to show an angle in standard position, and we also referred to the sine and cosine functions as . for the input (as well as to label the horizontal axis). . You know that the graphs of the sine and cosine functions have a pattern of hills and valleys that repeat. . (or ) on the interval looks like the graph on the interval . The graph below shows four repetitions of a pattern of length . is on the interval is one cycle. You know from graphing quadratic functions of the form that as you changed the value of a you changed the “width” of the graph. and see how changes to b will affect the graph. periodic, and if so, what is the period?

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. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. 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. Now let's looks at j(t) = sin(t – π/3):

4. Graphs of tan, cot, sec and csc by M. Bourne The graphs of tanx, cotx, secx and cscx are not as common as the sine and cosine curves that we met earlier in this chapter. However, they do occur in engineering and science problems. They are interesting curves because they have discontinuities. [For more on this topic, go to Continuous and Discontinuous Functions in an earlier chapter.] Recall from Trigonometric Functions, that tanx is defined as: tanx=​cos x​​sin x​​ For some values of x, the function cosx has value 0. When this happens, we have 0 in the denominator of the fraction and this means the fraction is undefined. The same thing happens with cotx, secx and cscx for different values of x. The Graph of y = tan x Sketch y = tan x. Solution As we saw above, tanx=​cosx​​sinx​​ This means the function will have a discontinuity where cos x = 0. x=…,−​2​​5π​​,−​2​​3π​​,−​2​​π​​,​2​​π​​,​2​​3π​​,​2​​5π​​,… It is very important to keep these values in mind when sketching this graph. Graph of y = tan x: Interactive Tangent Curve π 2π t

The Graphs of Sine and Cosine A sine wave, or sinusoid, is the graph of the sine function in trigonometry. A sinusoid is the name given to any curve that can be written in the form (A and B are positive). In addition to mathematics, sinusoidal functions occur in other fields of study such as science and engineering. Let's start with an investigation of the simpler graphs of y = A sin(Bx) and y = A cos(Bx). The value A affects the amplitude. The value B is the number of cycles it completes in an interval of from 0 to or 360º. . and the graph will be a horizontal stretching. and the graph will be a horizontal shrinking. This problem is a combination of dealing with the values of A and B. (there will be a horizontal stretch). (or 720º). This problem is also a combination of dealing with the values of A and B. (there will be a horizontal shrink). (or 120º). Look out for this problem.

Lesson HOW TO - Solve Trigonometric equations Introduction The solution of trigonometric equations is one topic that students have particular problems with. There are a few reasons for this: 1. there is usually a simplify part first that requires use of some TRIG identities. 2. there is the use of RADIANS rather than degrees, for which some students are not at ease with. 3. there is the repetitive aspect of TRIG functions that students find bewildering. All in all, a potentially daunting topic. To be honest though, there is nothing that you need fear here, so long as you take it methodically and slowly. Solving the TRIG Equation Of the 3 topics listed above, I am concentrating on part 3, here in this Lesson. First thing, when solving a TRIG equation, is to understand or accept that each of SINE, COSINE and TANGENT have 2 angles that will satisfy the given equation within any 360 degree range. If we have multiples, such as cos(2x), then we also multiply up the possible number of solutions. Quadrant 1 is the top right one. Done! 1.

Properties of The Six Trigonometric Functions The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x) are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points. Sine Function : f(x) = sin (x) Graph Domain: all real numbers Range: [-1 , 1] Period = 2pi x intercepts: x = k pi , where k is an integer. y intercepts: y = 0 maximum points: (pi/2 + 2 k pi , 1) , where k is an integer. minimum points: (3pi/2 + 2 k pi , -1) , where k is an integer. symmetry: since sin(-x) = - sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0 , 0). intervals of increase/decrease: over one period and from 0 to 2pi, sin (x) is increasing on the intervals (0 , pi/2) and (3pi/2 , 2pi), and decreasing on the interval (pi/2 , 3pi/2). Cosine Function : f(x) = cos (x) Graph Tangent Function : f(x) = tan (x) Graph Domain: all real numbers except pi/2 + k pi, k is an integer. Graph Graph Graph More on

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