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TF.02.6 - Linear Trigonometric Equations - Graphical Solutions

MCR3U - SantowskiMCR3U - Santowski

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(A) Review

We know what the graphs of the trigonometric We know what the graphs of the trigonometric functions look likefunctions look like

We know that when we algebraically solve an We know that when we algebraically solve an equation in the form of f(x) = 0, then we are trying equation in the form of f(x) = 0, then we are trying to find the roots/zeroes/x-interceptsto find the roots/zeroes/x-intercepts

So we should be able to solve trig equations by So we should be able to solve trig equations by graphing them and finding the x-interceptsgraphing them and finding the x-intercepts

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(B) Graphs of sin(x) and cos(x)

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(B) Graph of tan(x)

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(C) Examples

Solve the equation 3sin(x) – 2 = 0Solve the equation 3sin(x) – 2 = 0 We can set it up algebraically as sin(x) = 2/3 so x We can set it up algebraically as sin(x) = 2/3 so x

= sin= sin-1(2/3) giving us 41.8° (and the second angle (2/3) giving us 41.8° (and the second angle being 180° - 41.8° = 138.2°being 180° - 41.8° = 138.2°

Note that the ratio 2/3 is not one of our standard Note that the ratio 2/3 is not one of our standard ratios corresponding to our “standard” angles ratios corresponding to our “standard” angles (30,45,60), so we would use a calculator to (30,45,60), so we would use a calculator to actually find the related acute angle of 41.8°actually find the related acute angle of 41.8°

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(C) Examples We can now solve the equation 3sin(x) – 2 = 0 by graphing f(x) = 3sin(x) – 2 We can now solve the equation 3sin(x) – 2 = 0 by graphing f(x) = 3sin(x) – 2

and looking for the x-interceptsand looking for the x-intercepts

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(C) Examples

Notice that there are 2 solutions within the limited domain of 0° Notice that there are 2 solutions within the limited domain of 0° << << 360° 360°

However, if we expand our domain, then we get two new solutions for However, if we expand our domain, then we get two new solutions for every additional period we addevery additional period we add

The new solutions are related to the original solutions, as they The new solutions are related to the original solutions, as they represent the positive and negative co-terminal anglesrepresent the positive and negative co-terminal angles

We can determine their values by simply adding or subtracting We can determine their values by simply adding or subtracting multiples of 360° (the period of the given function)multiples of 360° (the period of the given function)

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(D) Example #2

Solve 4tan(2x) + 3 = 2Solve 4tan(2x) + 3 = 2 Again, we can set it up algebraically as tan(2x) = -1/4 and thus (2x) = Again, we can set it up algebraically as tan(2x) = -1/4 and thus (2x) =

tantan-1(-1/4) so x = ½ tan(-1/4) so x = ½ tan-1(-1/4)(-1/4) So thus x = ½ of 166° = 84°So thus x = ½ of 166° = 84° and x = ½ of 346° = 173°and x = ½ of 346° = 173°

To set it up graphically, we will make one minor change: we have two To set it up graphically, we will make one minor change: we have two graphing options graphing options we can graph f(x) = 4tan(2x) + 1 and find the x- we can graph f(x) = 4tan(2x) + 1 and find the x-intercepts OR we can graph f(x) = 4tan(2x) + 3 and find out where intercepts OR we can graph f(x) = 4tan(2x) + 3 and find out where f(x) = 2 f(x) = 2 to do this, we will simply graph y = 2 as a second equation to do this, we will simply graph y = 2 as a second equation and find out where the two graphs intersectand find out where the two graphs intersect

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(D) Example #2

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(D) Example #2

Notice that there are 2 solutions within the limited domain of 0° Notice that there are 2 solutions within the limited domain of 0° << << 180° 180° (NOTE: the period of a regular tan function is 180°)(NOTE: the period of a regular tan function is 180°)

However, if we expand our domain, then we get two new solutions for every However, if we expand our domain, then we get two new solutions for every additional period we addadditional period we add

The new solutions are related to the original solutions, as they represent the The new solutions are related to the original solutions, as they represent the positive and negative co-terminal anglespositive and negative co-terminal angles

We can determine their values by simply adding or subtracting multiples of We can determine their values by simply adding or subtracting multiples of 90° (the period of the given function 90° (the period of the given function the effect of the 2x in tan(2x) is a the effect of the 2x in tan(2x) is a horizontal compression by a factor of 2, so we have reduced the period by a horizontal compression by a factor of 2, so we have reduced the period by a factor of 2)factor of 2)

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(E) Internet Links

Go to the following applet from Go to the following applet from AnalyzeMath to see the connection between AnalyzeMath to see the connection between solving equations, triangles, and graphs:solving equations, triangles, and graphs:

Trigonometric Equations and The Unit CircTrigonometric Equations and The Unit Circle from le from AnalyzeMathAnalyzeMath

More on Trigonometric Equations - online More on Trigonometric Equations - online math lesson from math lesson from MathTVMathTV

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(F) Homework

Handouts from MHR and Nelson TextsHandouts from MHR and Nelson Texts

Nelson text, Chap 5.8, p474, Nelson text, Chap 5.8, p474, Q6,7,9,12,17,18Q6,7,9,12,17,18