Answer: [tex]\frac{11}{6}[/tex]
Step-by-step explanation:
30°-60°-90° triangle has corresponding sides of: b - b√3 - 2b
sec² [tex](\frac{\pi}{3})[/tex] = sec² 60°
sec² 60° = [tex]\frac{1}{cos^{2}60}[/tex]
= [tex](\frac{hypotenuse}{adjacent})^{2}[/tex]
= [tex](\frac{2b}{b\sqrt{3}})^{2}[/tex]
= [tex](\frac{2}{\sqrt{3}})^{2}[/tex]
= [tex]\frac{4}{3}[/tex]
45°-45°-90° triangle has corresponding sides of: a - a - a√2
sin² [tex](\frac{\pi}{4})[/tex] = sin² 45°
= [tex](\frac{opposite}{hypotenuse})^{2}[/tex]
= [tex](\frac{a}{a\sqrt{2}})^{2}[/tex]
= [tex](\frac{1}{\sqrt{2}})^{2}[/tex]
= [tex]\frac{1}{2}[/tex]
sec² [tex](\frac{\pi}{3})[/tex] + sin² [tex](\frac{\pi}{4})[/tex]
= [tex]\frac{4}{3}[/tex] + [tex]\frac{1}{2}[/tex]
= [tex]\frac{4}{3}(\frac{2}{2})[/tex] + [tex]\frac{1}{2}(\frac{3}{3})[/tex]
= [tex]\frac{8+3}{6}[/tex]
= [tex]\frac{11}{6}[/tex]
*********************************************************************************
(-12, 5)
- the x-coordinate is the adjacent side = -12
- the y-coordinate is the opposite side = 5
- Use Pythagorean Theorem to find the hypotenuse = 13
(-12)² + (5)² = (hypotenuse)²
144 + 25 = (hypotenuse)²
169 = (hypotenuse)²
13 = hypotenuse
sin θ = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{5}{13}[/tex]
cos θ = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]-\frac{12}{13}[/tex]
tan θ = [tex]\frac{opposite}{adjacent}[/tex] = [tex]-\frac{5}{12}[/tex]
csc θ = [tex]\frac{hypotenuse}{opposite}[/tex] = [tex]\frac{13}{5}[/tex]
sec θ = [tex]\frac{hypotenuse}{adjacent}[/tex] = [tex]-\frac{13}{12}[/tex]
cot θ = [tex]\frac{adjacent}{opposite}[/tex] = [tex]-\frac{12}{5}[/tex]
