Question

1. find the exact value of \\( \sin \theta \\), \\( \cos \theta \\), \\( \tan \theta \\) for an angle in standard position in quadrant ii where (-3, 4) lies on its terminal side. hint: draw your triangle, -3 & 4 are two of your triangle sides. \\( \sin = \\) \\( \cos = \\) \\( \tan = \\) 23. find the exact value of \\( \sin \theta \\), \\( \cos \theta \\), \\( \tan \theta \\) for an angle in standard position in quadrant iii where (-8, -15) lies on its terminal side. hint: draw your triangle, -8 & -15 are two of your triangle sides. \\( \sin = \\) \\( \cos = \\) \\( \tan = \\) 24. find the exact value of \\( \sin \theta \\), \\( \cos \theta \\), \\( \tan \theta \\) for an angle in standard position in quadrant i where (24,7) lies on its terminal side. hint: draw your triangle, 24 & 7 are two of your triangle sides. \\( \sin = \\) \\( \cos = \\)

Explanation:

Step1: Calculate the hypotenuse \(r\)

For a point \((x,y)\) on the terminal side of an angle \(\theta\), \(r=\sqrt{x^{2}+y^{2}}\).
For problem 22: \(x = - 3,y = 4\), then \(r=\sqrt{(-3)^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\)
For problem 23: \(x=-8,y = - 15\), then \(r=\sqrt{(-8)^{2}+(-15)^{2}}=\sqrt{64 + 225}=\sqrt{289}=17\)
For problem 24: \(x = 24,y = 7\), then \(r=\sqrt{24^{2}+7^{2}}=\sqrt{576+49}=\sqrt{625}=25\)

Step2: Use the trigonometric ratios

The trigonometric ratios are \(\sin\theta=\frac{y}{r}\), \(\cos\theta=\frac{x}{r}\), \(\tan\theta=\frac{y}{x}\)

Problem 22:

• \(\sin\theta=\frac{4}{5}\)
• \(\cos\theta=\frac{-3}{5}\)
• \(\tan\theta=\frac{4}{-3}=-\frac{4}{3}\)

Problem 23:

• \(\sin\theta=\frac{-15}{17}\)
• \(\cos\theta=\frac{-8}{17}\)
• \(\tan\theta=\frac{-15}{-8}=\frac{15}{8}\)

Problem 24:

• \(\sin\theta=\frac{7}{25}\)
• \(\cos\theta=\frac{24}{25}\)
• \(\tan\theta=\frac{7}{24}\)

Answer:

• Problem 22: \(\sin=\frac{4}{5}\), \(\cos=-\frac{3}{5}\), \(\tan=-\frac{4}{3}\)
• Problem 23: \(\sin=-\frac{15}{17}\), \(\cos=-\frac{8}{17}\), \(\tan=\frac{15}{8}\)
• Problem 24: \(\sin=\frac{7}{25}\), \(\cos=\frac{24}{25}\), \(\tan=\frac{7}{24}\)