Question

what is the approximate value of ( x ) in the diagram below? (hint: you will need to use one of the trigonometric ratios given in the table.)

( sin 43 ^ { circ } approx 0.682 )
( cos 43 ^ { circ } approx 0.731 )
( \tan 43 ^ { circ } approx 0.933 )

a. 36.94
b. 25.19
c. 19.74
d. 39.59
e. 28.94
f. 18.41

Explanation:

Step1: Identify the trigonometric ratio

In the right triangle, we have the side adjacent to the \(43^\circ\) angle as 27? Wait, no. Wait, the angle of \(43^\circ\), the side opposite? Wait, no, let's check the triangle. The right angle is at the bottom left, so the sides: the vertical side is 27, the angle at the bottom right is \(43^\circ\), and the hypotenuse is \(x\). Wait, the angle at the top is \(47^\circ\), and the angle at the bottom right is \(43^\circ\), so the vertical side (opposite to \(43^\circ\))? Wait, no. Wait, in a right triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Let's see: the angle \(43^\circ\), the adjacent side to \(43^\circ\) is the horizontal side, and the opposite side is the vertical side (length 27). Wait, no, the vertical side is adjacent to the \(47^\circ\) angle? Wait, maybe I got the angles wrong. Wait, the triangle has angles \(90^\circ\), \(47^\circ\), and \(43^\circ\) (since \(90 + 47+43=180\)). So the side of length 27 is adjacent to the \(43^\circ\) angle? Wait, no. Wait, the angle at the bottom right is \(43^\circ\), so the side opposite to \(43^\circ\) is the vertical side (length 27), and the hypotenuse is \(x\). Wait, no, \(\sin(43^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{27}{x}\)? Wait, no, wait: if the angle is \(43^\circ\), the opposite side is 27, then \(\sin(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{\sin(43^\circ)}\). Wait, but \(\sin(43^\circ)\approx0.682\), so \(x=\frac{27}{0.682}\approx39.59\)? Wait, no, that's not matching. Wait, maybe I mixed up the angle. Wait, the angle at the top is \(47^\circ\), so the side of length 27 is adjacent to the \(47^\circ\) angle, and the hypotenuse is \(x\). So \(\cos(47^\circ)=\frac{27}{x}\), but \(\cos(47^\circ)=\sin(43^\circ)\) (since \(\cos(90^\circ-\theta)=\sin\theta\)), so \(\sin(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{\sin(43^\circ)}\approx\frac{27}{0.682}\approx39.59\). Wait, but let's check again. Wait, maybe the angle is \(43^\circ\), and the adjacent side is 27, and the hypotenuse is \(x\). Then \(\cos(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{\cos(43^\circ)}\). \(\cos(43^\circ)\approx0.731\), so \(x=\frac{27}{0.731}\approx36.94\). Wait, now I'm confused. Wait, let's label the triangle: let's call the right angle \(C\), the top angle \(A = 47^\circ\), the bottom right angle \(B = 43^\circ\). Then side \(AC\) is vertical (length 27), side \(BC\) is horizontal, and side \(AB\) is hypotenuse \(x\). So angle at \(B\) is \(43^\circ\), so side \(AC\) (opposite to \(B\)) is 27, side \(BC\) is adjacent to \(B\), and side \(AB\) is hypotenuse. So \(\sin(B)=\frac{AC}{AB}\), so \(\sin(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{\sin(43^\circ)}\approx\frac{27}{0.682}\approx39.59\). But that's option D. Wait, but let's check the trigonometric ratios given. The table has \(\sin43\approx0.682\), \(\cos43\approx0.731\), \(\tan43\approx0.933\). So if we use \(\cos(47^\circ)=\sin(43^\circ)\approx0.682\), but maybe the side 27 is adjacent to \(43^\circ\). Wait, maybe I made a mistake in identifying the angle. Let's try again: the angle at the bottom right is \(43^\circ\), the side adjacent to \(43^\circ\) is the horizontal side, and the side opposite is the vertical side (27), and hypotenuse is \(x\). So \(\sin(43^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{27}{x}\), so \(x=\frac{27}{\sin(43^\circ)}\approx\frac{27}{0.682}\approx39.59\), which is option D. Alternatively, if the side 27 is adjacent to \(43^\circ\), then \(\cos(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{\cos(43^\circ)}\approx\frac{27}{0.731}\approx36.94\) (option A). Wait, now I'm confused. Wait, let's calculate both:

If \(x=\frac{27}{\sin(43^\circ)}\approx\frac{27}{0.682}\approx39.59\) (option D)

If \(x=\frac{27}{\cos(43^\circ)}\approx\frac{27}{0.731}\approx36.94\) (option A)

Wait, maybe the angle is \(47^\circ\). Let's check: angle \(47^\circ\), adjacent side 27, hypotenuse \(x\). So \(\cos(47^\circ)=\frac{27}{x}\), and \(\cos(47^\circ)=\sin(43^\circ)\approx0.682\), so \(x=\frac{27}{0.682}\approx39.59\) (option D). Alternatively, \(\sin(47^\circ)=\cos(43^\circ)\approx0.731\), so if the side 27 is opposite to \(47^\circ\), then \(\sin(47^\circ)=\frac{27}{x}\), so \(x=\frac{27}{0.731}\approx36.94\) (option A). Wait, now I need to figure out which side is which. Let's look at the triangle: the right angle is at the bottom left, so the vertical side is 27 (left side), the horizontal side is the base, and the hypotenuse is the top side (x). The angle at the top is \(47^\circ\), so the angle between the left side (27) and the hypotenuse (x) is \(47^\circ\). Therefore, the left side (27) is adjacent to the \(47^\circ\) angle, and the hypotenuse (x) is the hypotenuse. So \(\cos(47^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{27}{x}\). Since \(\cos(47^\circ)=\sin(43^\circ)\) (because \(\cos(90^\circ-\theta)=\sin\theta\), and \(90-47=43\)), so \(\sin(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{\sin(43^\circ)}\approx\frac{27}{0.682}\approx39.59\), which is option D. Wait, but let's check with \(\cos(43^\circ)\). If the angle at the bottom right is \(43^\circ\), then the left side (27) is opposite to \(43^\circ\), so \(\sin(43^\circ)=\frac{27}{x}\), so \(x=\frac{27}{0.682}\approx39.59\), which is option D. So that's the correct approach.

Step1: Identify the trigonometric ratio

In the right triangle, angle \(43^\circ\), opposite side = 27, hypotenuse = \(x\). So \(\sin(43^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{27}{x}\).

Step2: Solve for \(x\)

Rearrange the formula: \(x = \frac{27}{\sin(43^\circ)}\).

Substitute \(\sin(43^\circ) \approx 0.682\):

\(x = \frac{27}{0.682} \approx 39.59\).

Answer:

D. 39.59