`y = sin(pix) + cos(pix)` Find the x-intercepts of the graph.

Textbook Question

Chapter 5, 5.3 - Problem 46 - Precalculus (3rd Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the x intercepts of the graph, hence, you need to remember that the graph intercepts x axis at y = 0. Hence, you need to solve for x the equation `y= f(x) = 0` .

`sin pi*x + cos pi*x = 0`

You need to divide by `cos pi*x` , such that:

`(sin pi*x)/(cos pi*x) + 1 = 0`

`tan pi*x = -1 => pi*x = arctan (-1) + k*pi`

`pi*x = -pi/4 + k*pi`

Dividing by `pi` both sides, yields:

`x = -1/4 + k`

Hence, evaluating the x intercepts of the graph of the given function, yields `x = k - 1/4` , where` k in Z` .

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scisser | (Level 3) Honors

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Square the equation.

`sin^2(pix)+2sin(pix)cos(pix)+cos^2(pix)=0`

Use Pythagorean identities.

`sin^2x+cos^2x=1`

`2sin(pix)cos(pix)+1=0`

Use double angle formula for sin:

`sin(2x)=2sinxcosx`

`sin(2pix)=-1`

sine of -1 equals `3pi/2+2kpi`

`2pix=(3pi)/2+2kx`

`x=3/4+k`

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