Question

unit: trigonometry progress: the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. what are the values of sin x and tan y in the triangle below? triangle with right angle at z, xz=6, zy=3, xy=√34 options: sin x = 5/√34, tan y = 3/5; sin x = 5/√34, tan y = 5/3; sin x = 3/√34, tan y = 3/5; sin x = 3/√34, tan y = 5/3

Explanation:

Step1: Recall Trigonometric Ratios

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\).

For angle \(X\) in right - triangle \(XZY\) (right - angled at \(Z\)):
- The side opposite to angle \(X\) is \(YZ = 3\).
- The hypotenuse is \(XY=\sqrt{34}\).
So, \(\sin X=\frac{\text{opposite to }X}{\text{hypotenuse}}=\frac{YZ}{XY}=\frac{3}{\sqrt{34}}\) (rationalizing the denominator, we can also write it as \(\frac{3\sqrt{34}}{34}\), but the form \(\frac{3}{\sqrt{34}}\) is also correct as per the options).

For angle \(Y\) in right - triangle \(XZY\):
- The side opposite to angle \(Y\) is \(XZ = 6\)? Wait, no. Wait, \(XZ = 6\), \(YZ = 3\), \(XY=\sqrt{34}\). Wait, angle \(Y\): the opposite side to \(Y\) is \(XZ = 6\)? Wait, no, let's re - check. Wait, in right - triangle \(XZY\), right - angled at \(Z\), so:
- For angle \(Y\), the adjacent side is \(YZ = 3\) and the opposite side is \(XZ=6\)? Wait, no, wait the lengths: \(XZ = 6\), \(YZ = 3\), \(XY=\sqrt{34}\) (since \(6^{2}+3^{2}=36 + 9=45\)? Wait, no, \(6^{2}+3^{2}=36 + 9 = 45\), but \(\sqrt{34}\) squared is \(34\). Wait, there is a mistake here. Wait, maybe the lengths are \(XZ = 5\)? Wait, no, the given triangle has \(XZ = 6\), \(YZ = 3\), hypotenuse \(\sqrt{34}\)? Wait, \(6^{2}+3^{2}=36 + 9=45 eq34\). Wait, maybe the lengths are \(XZ = 5\), \(YZ = 3\), then \(5^{2}+3^{2}=25 + 9 = 34\), so \(XY=\sqrt{34}\). Oh, I see, maybe a typo in the diagram, \(XZ = 5\) (not 6). Let's correct that.

So, if \(XZ = 5\), \(YZ = 3\), \(XY=\sqrt{34}\) (since \(5^{2}+3^{2}=25 + 9=34\)):

For angle \(X\):
- Opposite side to \(X\) is \(YZ = 3\), hypotenuse \(XY=\sqrt{34}\), so \(\sin X=\frac{3}{\sqrt{34}}\) (or \(\frac{3\sqrt{34}}{34}\)).

For angle \(Y\):
- Opposite side to \(Y\) is \(XZ = 5\), adjacent side to \(Y\) is \(YZ = 3\). So, \(\tan Y=\frac{\text{opposite to }Y}{\text{adjacent to }Y}=\frac{XZ}{YZ}=\frac{5}{3}\)? Wait, no, wait:

Wait, no, let's start over. Let's define the right - triangle properly. Let's assume the right - triangle has legs \(a\) and \(b\) and hypotenuse \(c\), with \(a = 3\), \(b = 5\) (since \(3^{2}+5^{2}=9 + 25 = 34\)), so hypotenuse \(c=\sqrt{34}\). So, the right - angle is at \(Z\), so \(XZ = 5\), \(YZ = 3\), \(XY=\sqrt{34}\).

Now, angle \(X\):
- The side opposite to angle \(X\) is \(YZ = 3\), hypotenuse \(XY=\sqrt{34}\), so \(\sin X=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{3}{\sqrt{34}}\).

Angle \(Y\):
- The side opposite to angle \(Y\) is \(XZ = 5\), the side adjacent to angle \(Y\) is \(YZ = 3\), so \(\tan Y=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{3}\)? Wait, no, wait the options:

Wait, the options are:

1. \(\sin X=\frac{5}{\sqrt{34}},\tan Y=\frac{3}{5}\)
2. \(\sin X=\frac{5}{\sqrt{34}},\tan Y=\frac{5}{3}\)
3. \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{3}{5}\)
4. \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}\)

Wait, let's re - identify the triangle. Let's label the right - triangle: right - angled at \(Z\), so vertices \(X\), \(Z\), \(Y\) with \(Z\) the right - angle. So, \(XZ\) and \(YZ\) are the legs, \(XY\) is the hypotenuse.

If \(XZ = 5\), \(YZ = 3\), then:

- For angle \(X\):
- Opposite side: \(YZ = 3\)
- Hypotenuse: \(XY=\sqrt{34}\)
- So, \(\sin X=\frac{3}{\sqrt{34}}\)

- For angle \(Y\):
- Opposite side: \(XZ = 5\)
- Adjacent side: \(YZ = 3\)
- So, \(\tan Y=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{3}\)

Wait, but the fourth option (the last one) is \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}\) (the fourth option from the left: \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}\))

Wait, let's check the options again:

First option: \(\sin X=\frac{5}{\sqrt{34}},\tan Y=\frac{3}{5}\)

Second option: \(\sin X=\frac{5}{\sqrt{34}},\tan Y=\frac{5}{3}\)

Third option: \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{3}{5}\)

Fourth option: \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}\)

So, using the correct trigonometric ratios:

\(\sin X=\frac{\text{opposite to }X}{\text{hypotenuse}}=\frac{YZ}{XY}=\frac{3}{\sqrt{34}}\)

\(\tan Y=\frac{\text{opposite to }Y}{\text{adjacent to }Y}=\frac{XZ}{YZ}=\frac{5}{3}\) (assuming \(XZ = 5\), \(YZ = 3\))

So the correct option is the fourth one (the one with \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}\))

Step2: Match with Options

By comparing the calculated values of \(\sin X\) and \(\tan Y\) with the given options, we find that the option where \(\sin X=\frac{3}{\sqrt{34}}\) and \(\tan Y=\frac{5}{3}\) is the correct one.

Answer:

The option (the fourth one from the left) with \(\boldsymbol{\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}}\) (i.e., the last option among the four, written as \(\sin X=\frac{3}{\sqrt{34}},\tan Y=\frac{5}{3}\) in the options).