Question

1. for each problem below, let θ represent the angle between the positive x - axis and the segment from the origin through the given point. draw a sketch of the angle and its reference triangle. then find the exact 3rd side of the triangle, the exact side ratio values for sinθ and cosθ, the value of θ to the nearest whole degree, and its reference angle.

a) (-8, -15) b) (-2, 5)
3rd side length: 3rd side length:
sinθ = cosθ = sinθ = cosθ =
θ = reference angle = θ = reference angle =

1. identify an angle that is coterminal with the given angle.

a) 450° b) - 45° c) - 500°

Explanation:

8a) $(-8,-15)$

Step1: Calculate the length of the hypotenuse (3rd side)

Using the Pythagorean theorem \(r=\sqrt{x^{2}+y^{2}}\), where \(x = - 8\) and \(y=-15\).

$$r=\sqrt{(-8)^{2}+(-15)^{2}}=\sqrt{64 + 225}=\sqrt{289}=17$$

Step2: Calculate \(\sin\theta\) and \(\cos\theta\)

\(\sin\theta=\frac{y}{r}\), \(\cos\theta=\frac{x}{r}\). Since \(x=-8\), \(y = - 15\), \(r = 17\)
\(\sin\theta=\frac{-15}{17}\), \(\cos\theta=\frac{-8}{17}\)

Step3: Calculate \(\theta\)

\(\theta=\arctan(\frac{y}{x})+180^{\circ}\) (because the point \((-8,-15)\) is in the third - quadrant). \(\arctan(\frac{-15}{-8})=\arctan(\frac{15}{8})\approx62^{\circ}\), so \(\theta\approx180 + 62=242^{\circ}\)

Step4: Calculate the reference angle

The reference angle \(\theta_{r}=242^{\circ}-180^{\circ}=62^{\circ}\)

8b) \((-2,5)\)

Step1: Calculate the length of the hypotenuse (3rd side)

Using the Pythagorean theorem \(r=\sqrt{x^{2}+y^{2}}\), where \(x=-2\) and \(y = 5\)

$$r=\sqrt{(-2)^{2}+5^{2}}=\sqrt{4 + 25}=\sqrt{29}$$

Step2: Calculate \(\sin\theta\) and \(\cos\theta\)

\(\sin\theta=\frac{y}{r}=\frac{5}{\sqrt{29}}=\frac{5\sqrt{29}}{29}\), \(\cos\theta=\frac{x}{r}=\frac{-2}{\sqrt{29}}=-\frac{2\sqrt{29}}{29}\)

Step3: Calculate \(\theta\)

\(\theta=\arctan(\frac{y}{x})+180^{\circ}\) (because the point \((-2,5)\) is in the second - quadrant). \(\arctan(\frac{5}{-2})\approx - 68^{\circ}\), \(\theta\approx180^{\circ}-68^{\circ}=112^{\circ}\)

Step4: Calculate the reference angle

The reference angle \(\theta_{r}=180^{\circ}-112^{\circ}=68^{\circ}\)

9a) \(450^{\circ}\)

Step1: Find a coterminal angle

Coterminal angles are of the form \(\alpha=\beta + 360^{\circ}n\), where \(n\in\mathbb{Z}\). For \(\beta = 450^{\circ}\), if \(n=-1\)

$$450^{\circ}-360^{\circ}=90^{\circ}$$

9b) \(-45^{\circ}\)

Answer:

• 8a)
• 3rd side length: \(17\)
• \(\sin\theta=-\frac{15}{17}\), \(\cos\theta=-\frac{8}{17}\)
• \(\theta\approx242^{\circ}\), reference angle: \(62^{\circ}\)
• 8b)
• 3rd side length: \(\sqrt{29}\)
• \(\sin\theta=\frac{5\sqrt{29}}{29}\), \(\cos\theta=-\frac{2\sqrt{29}}{29}\)
• \(\theta\approx112^{\circ}\), reference angle: \(68^{\circ}\)
• 9a) A coterminal angle: \(90^{\circ}\)
• 9b) A coterminal angle: \(315^{\circ}\)
• 9c) A coterminal angle: \(220^{\circ}\)