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RoseWind [281]
3 years ago
6

If (5,-12) is a point on the terminal side of an angle θ, find the value of sin⁡θ, cos⁡θ, and tan⁡θ.

Mathematics
1 answer:
Bezzdna [24]3 years ago
5 0

Answer:

Given point (5,-12) falls into IV quadrant

<u>The right triangle with legs of 5 and -12, and hypotenuse is:</u>

  • √5² + (-12)² = √25+144 = √169 = 13

<u>Value of sin⁡θ, cos⁡θ, and tan⁡θ:</u>

  • sin⁡ θ = -12/13
  • cos ⁡θ = 5/13
  • tan⁡ θ = -12/5

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Segment EF: y = -x + 8

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\displaystyle\mathsf{\overline{BC}\:\: and\:\:\overline{EF}}

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\displaystyle\mathsf{Slope\:(m)\:=\:\frac{y_2 \:-\:y_1}{x_2 \:-\:x_1}}  }

Use the following coordinates from the given diagram:

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\displaystyle\mathsf{Slope\:(m)\:=\:\frac{y_2 \:-\:y_1}{x_2 \:-\:x_1}}\:=\:\frac{1\:-\:4}{1\:-\:(-2)}\:=\:\frac{-3}{1\:+\:2}\:=\:\frac{-3}{3}\:=\:-1}

<h2>Slope of Segment EF:</h2>

Similar to how we determined the slope of segment BC, we will use the coordinates of points E and F from ΔDEF to find its slope:

Point E:  (x₁, y₁) =  (4, 4)

Point F:  (x₂, y₂) = (6, 2)

Substitute these values into the slope formula:

\displaystyle\mathsf{Slope\:(m)\:=\:\frac{y_2 \:-\:y_1}{x_2 \:-\:x_1}}\:=\:\frac{2\:-\:4}{6\:-\:4}\:=\:\frac{-2}{2}\:=\:-1}

Our calculations show that segment BC and EF have the same slope of -1.  In geometry, we know that two nonvertical lines are <u>parallel</u> if and only if they have the same slope.  

Since segments BC and EF have the same slope, then it means that  \displaystyle\mathsf{\overline{BC}\:\: | |\:\:\overline{EF}}.

<h2>Slope-intercept form:</h2><h3><u>Segment BC:</u></h3>

The <u>y-intercept</u> is the point on the graph where it crosses the y-axis. Thus, it is the value of "y" when x = 0.

Using the slope of segment BC, m = -1, and the coordinates of point C, (1,  1), substitute these values into the <u>slope-intercept form</u> (y = mx + b) to solve for the y-intercept, <em>b. </em>

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Add 1 to both sides to isolate b:

1 + 1 = -1 + 1 + b

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Therefore, the linear equation in <u>slope-intercept form of segment BC</u> is:

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<h3><u /></h3><h3><u>Segment EF:</u></h3>

Using the slope of segment EF, <em>m</em> = -1, and the coordinates of point E, (4, 4), substitute these values into the <u>slope-intercept form</u> to solve for the y-intercept, <em>b. </em>

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