Question

applicationa=30 thinking30

1. solve for the unknowns in the diagrams below.a=4

a) solution
property of canstem education private school inc. 2013 - 2014
canstem education private school inc.
b) solution

Explanation:

Step1: Use trigonometric relations for part a

In the right - triangle in part a, we know one side and an angle. We use the sine function to find side \(c\). The sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 35^{\circ}\), the opposite side to the \(35^{\circ}\) angle is \(15\), and the hypotenuse is \(c\). So \(\sin35^{\circ}=\frac{15}{c}\).
\[c=\frac{15}{\sin35^{\circ}}\]
\[c=\frac{15}{0.5736}\approx26.15\]
We use the tangent function to find side \(BC\). \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan35^{\circ}=\frac{15}{BC}\), and \(BC = \frac{15}{\tan35^{\circ}}=\frac{15}{0.7002}\approx21.42\).

Step2: Use the Pythagorean theorem and trigonometric relations for part b

First, use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) to find \(x\). Here, \(a = 12\), \(c = 25\), so \(x=\sqrt{25^{2}-12^{2}}=\sqrt{(25 + 12)(25 - 12)}=\sqrt{37\times13}=\sqrt{481}\approx21.93\).
To find \(\theta\), we use the cosine function. \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos\theta=\frac{12}{25}=0.48\), and \(\theta=\cos^{-1}(0.48)\approx61.31^{\circ}\)

Answer:

a) \(c\approx26.15\), \(BC\approx21.42\)
b) \(x\approx21.93\), \(\theta\approx61.31^{\circ}\)