1. Problem: Find the value of $\theta$ if $\tan \theta = 1$. The tangent function equals 1 at angles where the opposite and adjacent sides are equal. The principal angle is $\theta = 45^\circ$. 2. Problem: Find $\sin \theta$ if the point $(12, -5)$ lies on the terminal side of $\theta$. Use the formula $\sin \theta = \frac{y}{r}$ where $r = \sqrt{x^2 + y^2}$. Calculate $r = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13$. Then $\sin \theta = \frac{-5}{13}$. 3. Problem: Given $\cos \theta = \frac{3}{5}$ and $\theta$ in the first quadrant, find $\sin \theta$. Use Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Calculate $\sin \theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$. 4. Problem: Find the cofunction of $\sec 14.9^\circ$. Cofunction identity: $\sec \theta = \csc (90^\circ - \theta)$. Calculate $90^\circ - 14.9^\circ = 75.1^\circ$ or $75^\circ 6' 0''$. So the cofunction is $\csc 75^\circ 6' 0''$. 5. Problem: Simplify $\sec x + \tan x$. Use identity $\sec x + \tan x = \frac{1}{\cos x} + \frac{\sin x}{\cos x} = \frac{1 + \sin x}{\cos x}$. 6. Problem: Simplify $\sec x - \cos x$. Rewrite $\sec x = \frac{1}{\cos x}$, so $\sec x - \cos x = \frac{1}{\cos x} - \cos x = \frac{1 - \cos^2 x}{\cos x} = \frac{\sin^2 x}{\cos x} = \sin x \tan x$. 7. Problem: Find $\sin (\alpha + \beta)$. Use sum formula: $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$. 8. Problem: Given $\tan x = \frac{1}{2}$ and $\tan y = \frac{1}{3}$, find $\tan (x + y)$. Use formula $\tan (x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2} \cdot \frac{1}{3}} = \frac{\frac{3}{6} + \frac{2}{6}}{1 - \frac{1}{6}} = \frac{\frac{5}{6}}{\frac{5}{6}} = 1$. 9. Problem: Identify the identity $\sin^2 u + \cos^2 u = 1$. This is the Pythagorean identity. 10. Problem: Given $\sin \theta = 0.28$, find possible $\cos \theta$. Use $\cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - 0.0784} = \pm \sqrt{0.9216} = \pm 0.96$. Possible values are $0.96$ or $-0.96$. 11. Problem: Find $\cos (x + y)$. Use formula $\cos (x + y) = \cos x \cos y - \sin x \sin y$. 12. Problem: Simplify $\cos^2 x - \sin^2 x$. Use double angle identity $\cos 2x = \cos^2 x - \sin^2 x$. 13. Problem: Find amplitude of $y = 3 \sin x$. Amplitude is the coefficient of $\sin x$, so amplitude = 3. 14. Problem: Find period of $y = \sin (4x - \frac{\pi}{2})$. Period of $\sin bx$ is $\frac{2\pi}{b}$. Here $b=4$, so period = $\frac{2\pi}{4} = \frac{\pi}{2}$. 15. Problem: Find shift of $y = \cos (x - \frac{\pi}{3})$. Shift is $\frac{\pi}{3}$ units to the right. 16. Problem: For $y = 5 \cos (3x) - 4$, find amplitude, period, midline. Amplitude = 5, period = $\frac{2\pi}{3}$, midline = $-4$. 17. Problem: For $y = 3 \sin (2x) + 4$, find vertical stretch, period, vertical shift. Vertical stretch by 3, period = $\frac{2\pi}{2} = \pi$, shifted up 4 units. 18. Problem: A plane is determined by I. any three non-collinear points II. a line and a point III. two intersecting lines IV. a line and a point not on it Correct answers: I and III. 19. Problem: What is formed by the intersection of two planes? Answer: a line.