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Rashid [163]
3 years ago
8

find the values of the six trigonometric functions for angle theta in standard position if a point with the coordinates (1, -8)

lies on its terminal side
Mathematics
1 answer:
frutty [35]3 years ago
8 0

Answer:

cosФ = \frac{1}{\sqrt{65}} , sinФ = -\frac{8}{\sqrt{65}} , tanФ = -8, secФ = \sqrt{65} , cscФ = -\frac{\sqrt{65}}{8} , cotФ = -\frac{1}{8}

Step-by-step explanation:

If a point (x, y) lies on the terminal side of angle Ф in standard position, then the six trigonometry functions are:

  1. cosФ = \frac{x}{r}
  2. sinФ = \frac{y}{r}
  3. tanФ = \frac{y}{x}
  4. secФ = \frac{r}{x}
  5. cscФ = \frac{r}{y}
  6. cotФ = \frac{x}{y}
  • Where r = \sqrt{x^{2}+y^{2} } (the length of the terminal side from the origin to point (x, y)
  • You should find the quadrant of (x, y) to adjust the sign of each function

∵ Point (1, -8) lies on the terminal side of angle Ф in standard position

∵ x is positive and y is negative

→ That means the point lies on the 4th quadrant

∴ Angle Ф is on the 4th quadrant

∵ In the 4th quadrant cosФ and secФ only have positive values

∴ sinФ, secФ, tanФ, and cotФ have negative values

→ let us find r

∵ r = \sqrt{x^{2}+y^{2} }

∵ x = 1 and y = -8

∴ r = \sqrt{x} \sqrt{(1)^{2}+(-8)^{2}}=\sqrt{1+64}=\sqrt{65}

→ Use the rules above to find the six trigonometric functions of Ф

∵ cosФ = \frac{x}{r}

∴ cosФ = \frac{1}{\sqrt{65}}

∵ sinФ = \frac{y}{r}

∴ sinФ = -\frac{8}{\sqrt{65}}

∵ tanФ = \frac{y}{x}

∴ tanФ = -\frac{8}{1} = -8

∵ secФ = \frac{r}{x}

∴ secФ = \frac{\sqrt{65}}{1} = \sqrt{65}

∵ cscФ = \frac{r}{y}

∴ cscФ = -\frac{\sqrt{65}}{8}

∵ cotФ = \frac{x}{y}

∴ cotФ = -\frac{1}{8}    

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Intervals are not my strongest suit when it comes to prepping for my act. I could use a little help, and advice if it isn’t too
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Step-by-step explanation:

Start from the left side.

The arrow on the left side shows that the graph continues increasing forever as x decreases. Beginning at the highest value of y you see on the left side at point (-6, 5), as x increases, moving right along the x-axis, the y-values decrease. When you get to x = -4, y is at its lowest value which is 1.

Starting just to the right of x = -4, the y-values begin to increase from the lowest y-value of 1. The arrowhead at the right side top shows that the curve continues increasing forever to infinity.

The values of x for which y increases are all values greater than -4 and all the way to positive infinity.

This function is increasing for x > -4.

Now we need to write x > -4 in interval notation.

In interval notation, use a curved parenthesis to mean a number that is not included. The interval starts at -4, so it starts as

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Then you write a comma to separate from the value where the interval ends.

(-4,

The interval in x of increasing values goes to positive infinity, so now you write the infinity symbol, and you close the interval with a curved parenthesis. By convention, infinity always gets a curved parenthesis. Also, I'll use oo for infinity below.

(-4, oo)

Answer: F. (-4, oo)

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