Question

4.
angle (θ)
opposite
adjacent
hypotenuse
trig function

Explanation:

Step1: Identify the sides relative to the angle

The angle \(\theta = 50.1^\circ\) is at vertex \(A\). In the right - triangle \(ABC\) (right - angled at \(C\)):

• The side opposite to \(\theta\) is \(BC\) (length \(x\)).
• The side adjacent to \(\theta\) is \(AC\) with length \(5\).
• The hypotenuse is \(AB\).

We can use the tangent function since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\theta = 50.1^\circ\), \(\tan(50.1^\circ)=\frac{x}{5}\).

Step2: Solve for \(x\)

We know that \(\tan(50.1^\circ)\approx1.199\) (using a calculator to find the tangent of \(50.1^\circ\)).
From \(\tan(50.1^\circ)=\frac{x}{5}\), we can solve for \(x\) by multiplying both sides of the equation by \(5\):
\(x = 5\times\tan(50.1^\circ)\)
Substitute \(\tan(50.1^\circ)\approx1.199\) into the equation:
\(x=5\times1.199 = 5.995\approx6\)

Answer:

If we are finding the length of \(x\) (the opposite side), the value of \(x\) is approximately \(6\). For the table:

• Angle \((\theta)\): \(50.1^\circ\)
• Opposite: \(x\) (or approximately \(6\))
• Adjacent: \(5\)
• Hypotenuse: \(AB\) (can be found using Pythagoras \(AB=\sqrt{5^{2}+x^{2}}\), if \(x = 6\), \(AB=\sqrt{25 + 36}=\sqrt{61}\approx7.81\))
• Trig Function: \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) (used to find \(x\))