Question

individual work #4 - trigonometric functions math 10 1st semester
iv. for the triangles shown, determine the missing side lengths. then, evaluate the trigonometric functions. 10 pts

Explanation:

Left Triangle:

Step1: Find \(AB\) and \(AC\)

Since it is a right - isosceles triangle (\(AB = AC\)) and by Pythagoras' theorem \(AB^{2}+AC^{2}=(7\sqrt{2})^{2}\). Let \(AB = AC = x\), then \(2x^{2}=98\), \(x^{2}=49\), \(x = 7\). So \(AB=7\), \(AC = 7\).

Step2: Calculate \(\csc\theta\)

\(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}\). In \(\triangle ABC\), \(\theta\) has opposite side \(AC = 7\) and hypotenuse \(BC=7\sqrt{2}\). So \(\csc\theta=\frac{7\sqrt{2}}{7}=\sqrt{2}\).

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). The adjacent side to \(\theta\) is \(AB = 7\) and hypotenuse \(BC = 7\sqrt{2}\). So \(\cos\theta=\frac{7}{7\sqrt{2}}=\frac{\sqrt{2}}{2}\).

Step4: Calculate \(\cot\theta\)

\(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\). Adjacent side \(AB = 7\) and opposite side \(AC = 7\). So \(\cot\theta=\frac{7}{7}=1\).

Right Triangle:

Step1: Find \(XY\)

By Pythagoras' theorem \(XY=\sqrt{YZ^{2}-XZ^{2}}\). Given \(YZ = 13\) and \(XZ = 5\), then \(XY=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\).

Step2: Calculate \(\cos\theta\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Adjacent side \(XY = 12\) and hypotenuse \(YZ = 13\). So \(\cos\theta=\frac{12}{13}\).

Step3: Calculate \(\sin\theta\)

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Opposite side \(XZ = 5\) and hypotenuse \(YZ = 13\). So \(\sin\theta=\frac{5}{13}\).

Step4: Calculate \(\sec\theta\)

\(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}\). Hypotenuse \(YZ = 13\) and adjacent side \(XY = 12\). So \(\sec\theta=\frac{13}{12}\).

Step5: Calculate \(\tan\theta\)

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Opposite side \(XZ = 5\) and adjacent side \(XY = 12\). So \(\tan\theta=\frac{5}{12}\).

Answer:

\(\overline{AB}\)	\(\overline{AC}\)	\(\csc\theta\)	\(\cos\theta\)	\(\cot\theta\)

\(\overline{XY}\)	\(\cos\theta\)	\(\sin\theta\)	\(\sec\theta\)	\(\tan\theta\)