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Find the values of the six trigonometric functions of an angle in standard position if the point with coordinates (12, 5) lies on its terminal side.

Respuesta :

calculista

Answer:

Part 1) [tex]sin(\beta)=\frac{5}{13}[/tex]

Part 2) [tex]cos(\beta)=\frac{12}{13}[/tex]

Part 3) [tex]tan(\beta)=\frac{5}{12}[/tex]

Part 4) [tex]cot(\beta)=\frac{12}{5}[/tex]

Part 5) [tex]sec(\beta)=\frac{13}{12}[/tex]

Part 6) [tex]csc(\beta)=\frac{13}{5}[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

In the right triangle ABC

we have

[tex]AC=12\ units, BC=5\ units[/tex]

Applying the Pythagoras Theorem

Find the length side AB (hypotenuse)

[tex]AB^{2}=AC^{2}+BC^{2}[/tex]

substitute the values

[tex]AB^{2}=12^{2}+5^{2}[/tex]

[tex]AB^{2}=169}[/tex]

[tex]AB=13\ units[/tex]

Part 1) Find the [tex]sin(\beta)[/tex]

[tex]sin(\beta)=\frac{BC}{AB}[/tex]

substitute the values

[tex]sin(\beta)=\frac{5}{13}[/tex]

Part 2) Find the [tex]cos(\beta)[/tex]

[tex]cos(\beta)=\frac{AC}{AB}[/tex]

substitute the values

[tex]cos(\beta)=\frac{12}{13}[/tex]

Part 3) Find the [tex]tan(\beta)[/tex]

[tex]tan(\beta)=\frac{BC}{AC}[/tex]

substitute the values

[tex]tan(\beta)=\frac{5}{12}[/tex]

Part 4) Find the [tex]cot(\beta)[/tex]

[tex]cot(\beta)=\frac{AC}{BC}[/tex]

substitute the values

[tex]cotan(\beta)=\frac{12}{5}[/tex]

Part 5) Find the [tex]sec(\beta)[/tex]

[tex]sec(\beta)=\frac{1}{cos(\beta)}[/tex]

substitute the values

[tex]sec(\beta)=\frac{13}{12}[/tex]

Part 6) Find the [tex]csc(\beta)[/tex]

[tex]csc(\beta)=\frac{1}{sin(\beta)}[/tex]

substitute the values

[tex]csc(\beta)=\frac{13}{5}[/tex]

Ver imagen calculista

There are 6 functions of the angle which is known as trigonometry function.

The values of the six trigonometric functions of an angle in standard position if the point with coordinates (12, 5) lies on its terminal side is,

[tex]\sin \theta =\dfrac{5}{13}[/tex]

[tex]\cos \theta =\dfrac{12}{13}[/tex]

[tex]\tan \theta =\dfrac{5}{12}[/tex]

[tex]\sec \theta=\dfrac{13}{12}[/tex]

[tex]\ cosec \theta=\dfrac{13}{5}[/tex]

[tex]\cot \theta =\dfrac{12}{5}[/tex]

What is trigonometric function?

There are 6 functions of the angle which is known as trigonometry function. This six angles are sine, cosine, tangent, secant, co secant and cotangent.

Given information-

The coordinates of the triangle point is (12,5).

In the attached figure the triangle [tex]XYZ[/tex] is shown. In this right angle triangle,

[tex]XY=12\\YZ=5\\XZ=\sqrt{(XY)^2+(YZ)^2} \\XZ=\sqrt{(12)^2+(5)^2}\\ XZ=\sqrt{(144+25}\\XZ=\sqrt{(169}\\XZ=13[/tex]

The six trigonometric functions-

Suppose the [tex]\angle YXZ[/tex] is [tex]\theta[/tex]. Thus,

  • a) SIne angle is the ratio of opposite side to the hypotenuse side.

           [tex]\sin \theta=\dfrac{YZ}{XZ} =\dfrac{5}{13}[/tex]

  •  b) Cosine angle is the ratio of adjacent side to the hypotenuse side.

           [tex]\cos \theta=\dfrac{XY}{YZ} =\dfrac{12}{13}[/tex]

  • c) Tangent angle is the ratio of opposite side to the adjacent side.

           [tex]\tan \theta=\dfrac{YZ}{AY} =\dfrac{5}{12}[/tex]

  • d) Secant angle is the ratio of hypotenuse side to the adjacent side.

           [tex]\sec \theta=\dfrac{XZ}{XY} =\dfrac{13}{12}[/tex]

  • e) Co secant angle is the ratio of hypotenuse side to the opposite side.

           [tex]\ cosec \theta=\dfrac{XZ}{XY} =\dfrac{13}{5}[/tex]

  • f)Co tangent  angle is the ratio of adjacent side to the opposite side.

           [tex]\cot \theta=\dfrac{XY}{YZ} =\dfrac{12}{5}[/tex]

Hence the values of the six trigonometric functions of an angle in standard position if the point with coordinates (12, 5) lies on its terminal side is,

[tex]\sin \theta =\dfrac{5}{13}[/tex]

[tex]\cos \theta =\dfrac{12}{13}[/tex]

[tex]\tan \theta =\dfrac{5}{12}[/tex]

[tex]\sec \theta=\dfrac{13}{12}[/tex]

[tex]\ cosec \theta=\dfrac{13}{5}[/tex]

[tex]\cot \theta =\dfrac{12}{5}[/tex]

Learn more about the trigonometry function here;

https://brainly.com/question/6904750

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