📏 trigonometry

1. **State the problem:** We have a right triangle with a right angle at the bottom-left corner, an angle of 39° at the top-left corner, the horizontal leg adjacent to the 39° angl

1. **State the problem:** We have a right triangle with a right angle at the bottom-left, an angle of 39° at the top-left, a horizontal leg of length 5 ft, and a vertical leg of le

1. The problem asks to determine the amplitude of a sinusoidal graph. 2. The amplitude of a sinusoidal function is the distance from the centerline (midline) to a peak (maximum) or

1. The problem asks to determine the amplitude of a sinusoidal graph. 2. The amplitude of a sinusoidal function is the distance from the midline (center) to a peak or trough.

1. **State the problem:** We have a straight line P–S–R with PS = 8.4 cm and a right angle at S (angle PSQ = 90°). Given angles QPS = 38° and SQR = 44°, we need to find the length

1. **State the problem:** Marion wants to find $\cos \frac{\pi}{12}$ given that $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$.\n\n2. **Recall the formula:** We can use the half-angle f

1. **State the problem:** Evaluate $\tan\left(\frac{\pi}{4} + \pi\right)$. 2. **Recall the tangent addition formula:**

1. Problem: Write the trigonometric ratios for the given triangles. **a. tan A in triangle ABC** (right angle at C, sides 28 cm, 24 cm, 10 cm)

1. **State the problem:** Verify the trigonometric identity $$\frac{\csc \theta \cdot \tan \theta}{\sec \theta} = 1$$ and show the steps to prove it. 2. **Recall the definitions an

1. **State the problem:** Prove that $$\frac{\tan y}{\sin y} = \sec y$$. 2. **Recall definitions:**

1. The problem asks for the domain of the function $y = \cos(x)$.\n\n2. The domain of a function is the set of all possible input values ($x$) for which the function is defined.\n\

1. **State the problem:** Solve the equation $\tan 4x = -1$ for $x$. 2. **Recall the formula and properties:** The tangent function satisfies $\tan \theta = -1$ at angles where $\t

1. **Stating the problem:** Solve the equation $50 \tan x = 1$ for $x$. 2. **Rewrite the equation:** Divide both sides by 50 to isolate $\tan x$:

1. **Problem Statement:** (a) Prove that $$\tan^2\theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$$ provided that $$1 + \cos 2\theta \neq 0$$.

1. **Stating the problem:** We have a right triangle with a vertical side of length 5.2 m and an angle of 23° adjacent to the base $x$. We want to find the length of the base $x$.

1. The problem asks to find the sine and cosine of the angle $\alpha$ given the point $(2,4)$ on its terminal side. 2. Recall that for a point $(x,y)$ on the terminal side of an an

1. **Problem Statement:** We have a right triangle with side lengths 7, 24, and 25. We need to find $\sin B$, $\tan B$, and $\cos B$ where angle $B$ is at the bottom-right vertex.

1. **State the problem:** We have a right triangle with sides 9, 40, and 41. We need to find $\tan X$, $\cos X$, and $\sin X$ where $X$ is the angle at vertex $X$. 2. **Identify si

1. The problem asks to estimate the angle of a green ray on the unit circle that starts at the origin and points to a position in Quadrant III, slightly below the negative x-axis,

1. The problem asks to estimate the rotation angle in degrees within 10 degrees for a ray in Quadrant II starting from the positive x-axis. 2. Recall that the coordinate plane is d

1. **State the problem:** We need to write the equation of a sinusoidal function given its graph characteristics. 2. **Identify key features from the graph:**