Question

the terminal ray of an angle θ intersects the unit circle as shown below. the slope of the terminal ray is 5.671. use that and/or the given coordinate of the point of intersection to calculate sin θ rounded to three decimal places, if necessary.

(-0.174, y)

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Explanation:

Step1: Recall unit circle equation

The equation of the unit circle is \(x^2 + y^2 = 1\). We know the x - coordinate of the point on the unit circle is \(x=- 0.174\).

Step2: Substitute x into the equation

Substitute \(x = - 0.174\) into the equation \(x^2+y^2 = 1\):
\[ \begin{align*} (-0.174)^2+y^2&=1\\ 0.030276 + y^2&=1\\ y^2&=1 - 0.030276\\ y^2&=0.969724 \end{align*} \]
Since the point \((-0.174,y)\) is in the fourth quadrant (from the diagram, the terminal ray is in the lower part, so y is negative), we take the negative square root:
\(y=-\sqrt{0.969724}\approx - 0.9847\approx - 0.985\)
But we know that for a point \((x,y)\) on the unit circle, \(\sin\theta=y\). Also, we can use the slope. The slope \(m = 5.671=\frac{y - 0}{x - 0}=\frac{y}{x}\), so \(y=m\times x\). Substitute \(m = 5.671\) and \(x=-0.174\):
\(y=5.671\times(-0.174)=- 5.671\times0.174\)
Calculate \(5.671\times0.174\):
\(5\times0.174 = 0.87\), \(0.671\times0.174=0.116754\), so \(5.671\times0.174=0.87 + 0.116754=0.986754\), then \(y=- 0.986754\approx - 0.987\) (Wait, there is a discrepancy because of the unit circle. Wait, the unit circle has radius 1, so \(x^2 + y^2=1\) must hold. Let's recalculate with \(x=-0.174\):

\(x^{2}=(-0.174)^{2}=0.030276\)

\(y^{2}=1 - 0.030276 = 0.969724\)

\(y=-\sqrt{0.969724}\approx - 0.9847\approx - 0.985\). But the slope is given as 5.671. Let's check the slope formula: slope between \((0,0)\) and \((x,y)\) is \(\frac{y - 0}{x - 0}=\frac{y}{x}\). So \(y = m\times x\). If \(m = 5.671\) and \(x=-0.174\), then \(y=5.671\times(-0.174)=- 5.671\times0.174\)

Calculate \(5.671\times0.174\):

\(5.671\times0.174=(5 + 0.6+0.07 + 0.001)\times0.174\)

\(5\times0.174 = 0.87\)

\(0.6\times0.174 = 0.1044\)

\(0.07\times0.174 = 0.01218\)

\(0.001\times0.174 = 0.000174\)

Sum: \(0.87+0.1044 + 0.01218+0.000174=0.986754\)

So \(y=- 0.986754\approx - 0.987\). But the unit circle equation: \(x^{2}+y^{2}=(-0.174)^{2}+(-0.986754)^{2}=0.030276 + 0.9737\approx1.003976\), which is close to 1 (due to rounding). So we can use the fact that on the unit circle, \(\sin\theta=y\), and from the slope \(y = m\times x\). So \(\sin\theta=y=5.671\times(-0.174)\approx - 0.987\) (or from the unit circle, \(y =-\sqrt{1 - x^{2}}=-\sqrt{1 - 0.030276}=-\sqrt{0.969724}\approx - 0.9847\approx - 0.985\). But since the slope is given, we should use the slope. Wait, maybe the diagram shows the angle in the fourth quadrant? Wait, no, the point is \((-0.174,y)\), x is negative, y is negative, so it's in the third quadrant. Wait, x negative, y negative: third quadrant. Then slope is positive? Wait, slope is \(\frac{y}{x}\), if x is negative and y is negative, slope is positive, which matches 5.671. So let's recast:

We know that for a point \((x,y)\) on the unit circle, \(x^{2}+y^{2}=1\), and \(m=\frac{y}{x}\Rightarrow y = m x\). Substitute into the unit circle equation:

\(x^{2}+(m x)^{2}=1\)

\(x^{2}(1 + m^{2})=1\)

\(x^{2}=\frac{1}{1 + m^{2}}\)

\(x=\pm\frac{1}{\sqrt{1 + m^{2}}}\)

Since x is negative (from the point \((-0.174,y)\)), \(x =-\frac{1}{\sqrt{1 + m^{2}}}\)

Then \(y=m x=-\frac{m}{\sqrt{1 + m^{2}}}\)

And \(\sin\theta=y=-\frac{m}{\sqrt{1 + m^{2}}}\)

Substitute \(m = 5.671\):

First, calculate \(1 + m^{2}=1+5.671^{2}=1 + 32.160241=33.160241\)

\(\sqrt{33.160241}\approx5.7585\)

Then \(y=-\frac{5.671}{5.7585}\approx - 0.9848\approx - 0.985\)

So \(\sin\theta=y\approx - 0.985\) (or - 0.987 depending on the method, but the unit circle method is more accurate as the point is on the unit circle). Let's use the unit circle equation:

\(x=-0.174\), so \(y^{2}=1 - (-0.174)^{2}=1 - 0.030276 = 0.969724\)

\(y=-\sqrt{0.969724}\approx - 0.9847\approx - 0.985\)

So \(\sin\theta=y\approx - 0.985\)

Answer:

\(\sin\theta\approx - 0.985\) (or - 0.987 if using slope directly, but the unit circle gives a more accurate value considering the point is on the unit circle. The difference is due to possible rounding in the slope value. If we use the given x - coordinate \(-0.174\) and unit circle: \(y =-\sqrt{1 - (-0.174)^2}=-\sqrt{1 - 0.030276}=-\sqrt{0.969724}\approx - 0.9847\approx - 0.985\))