Express x and y in terms of trigonometric ratios, express your answer in terms of theta only.

here you'll just be using the SOH CAH TOA thing (or that's the way i learned it), so

sin = opposite/hypotenuse
cos = adjacent/hypotenuse
tan = opposite/adjacent

so x, in relation to theta, is the OPPOSITE of the angle and ADJACENT to 6. because you're using opp and adj, you'll use tangent.

tan(theta) = (x/6)

but, because the answer calls for x to be isolated, it'll look like:

x = 6 tan (theta)

all you do is get x alone.

y is the HYPOTENUSE of the right triangle, so you can use either sine or cosine with it. I'll use cos, so it'll be using 6 (the adjacent) and the hypotenuse, aka Y

cos(theta) = (6/y)

then get y alone

y cos (theta) = 6
y = (6/costheta)

AkhileshT AkhileshT · Answer 2 · 2019-09-10T08:23:16+02:00

The value of [tex]x[/tex] and [tex]y[/tex] can be expressed in terms of trigonometric ratios as [tex]\boxed{x=6\text{ tan}(\theta)\ \text{and }y=6\text{ sec}(\theta)}[/tex]

Further explanation:

Trigonometric functions are the real functions. In any right angle triangle there are six trigonometric functions sine, cosine, tangent, secant, cosecant, cotangent.

Given:

The right angle triangle has the sides [tex]x,y\text{ and }6[/tex].

Calculation:

From the attached Figure 1, we can see that the triangle is the right angle triangle.

We can apply all trigonometric ratios here.

The length of the adjacent side to the angle [tex]\theta[/tex] is [tex]6\text{ cm}[/tex] and the length of the opposite side to the angle [tex]\theta[/tex] is the [tex]x[/tex] and the hypotenuse is [tex]y[/tex].

The value of [tex]\text{tan}(\theta)[/tex] is the ratio of the side which is opposite to the angle [tex]\theta[/tex] and the adjacent side to the angle [tex]\theta[/tex].

The value of [tex]\text{tan}(\theta)[/tex] can be mathematically expressed as follows:

[tex]\boxed{\text{tan}(\theta)=\dfrac{\text{opposite side to angle}(\theta)}{\text{adjacent side to angle}(\theta)}}[/tex]

Therefore, the value of [tex]\text{tan}(\theta)[/tex] can be evaluated as follows:

[tex]\boxed{\text{tan}(\theta)=\dfrac{x}{6}}[/tex]

Further solve the above equation as follows:

[tex]x=6\text{ tan}(\theta)[/tex]

Therefore, the value of [tex]x[/tex] in terms of trigonometric ratios is [tex]x=6\text{ tan}(\theta)[/tex].

The value of [tex]\text{cos}(\theta)[/tex] is the ratio of the side which is adjacent to the angle [tex]\theta[/tex] and the hypotenuse of the right angle triangle.

The value of [tex]\text{cos}(\theta)[/tex] can be mathematically expressed as follows:

[tex]\boxed{\text{cos}(\theta)=\dfrac{\text{adjacent side to angle}\ \theta}{\text{hypotenuse}}}[/tex]

Therefore, the value of [tex]\text{cos}\theta[/tex] can be evaluated as follows:

[tex]\boxed{\text{cos}\theta=\dfrac{6}{y}}[/tex]

Further solve the above equation as follows:

[tex]\begin{aligned}\text{cos}(\theta)&=\dfrac{6}{y}\\y&=\dfrac{6}{\text{cos}(\theta)}\\y&=6\text{ sec}(\theta)\end{aligned}[/tex]

Therefore, the value of [tex]y[/tex] in terms of trigonometric ratios is [tex]y=\text{sec}(\theta)[/tex].

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Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine, cosine, cotangent, tangent, cosecant, secant, theta, right angle triangle, adjacent side opposite side, hypotenuse, base, perpendicular.