📏 trigonometry

1. **State the problem:** Prove that $$\frac{\tan \theta}{1+\sec \theta} + \frac{1+\sec \theta}{\tan \theta} = 2 \csc \theta$$. 2. **Recall definitions and identities:**

1. **State the problem:** We need to analyze and understand the function $$f(x) = 2 \sin\left(\frac{\pi}{4} x\right) + 3$$ and sketch its graph. 2. **Formula and important rules:**

1. **Problem Statement:** Given the sine curve $y = \sin x^\circ$ for $0 \leq x \leq 360$, and knowing that $\sin 30^\circ = \frac{1}{2}$, find:

1. The problem states that the curve is $y = \sin x^\circ$ and point A is where the curve returns to the x-axis after rising to a maximum height of 2 units. 2. Normally, the sine f

1. **State the problem:** Solve the equation $$\tan x + 1 = -\sqrt{3} - \sqrt{3} \cot x$$ for $$x$$ in the interval $$[0, 2\pi)$$. 2. **Rewrite the equation:** Recall that $$\cot x

1. **State the problem:** Solve the equation $$\sec^2 \theta = 2 \tan \theta + 16$$ for $$\theta$$ in the interval $$[0^\circ, 360^\circ)$$. 2. **Recall the identity:** $$\sec^2 \t

1. **Problem statement:** Two planes are flying at an altitude of 2.5 miles approaching O’Hare Airport from opposite directions. Flight 104 sees the tower at an angle of depression

1. **State the problem:** Two planes leave JFK airport at different times and directions. We want to find how far apart they are 40 minutes after the second plane leaves.

1. **State the problem:** We have a lighthouse of height $h$ and two points $A$ and $B$ on the horizontal line from the base of the lighthouse. The horizontal distance from $A$ to

1. **State the problem:** We have a lighthouse beacon 141 feet above water. From point A, the angle of elevation to the beacon is 6°. From point B, closer to the lighthouse, the an

1. **State the problem:** Given a triangle with \(m\angle B = 19^\circ\), side \(a = 30\) yd opposite \(\angle A\), and side \(b = 12\) yd opposite \(\angle B\), find the remaining

1. **Problem 1:** Given $m\angle B = 14^\circ$, side $a = 30$ yd, and side $b = 12$ yd, solve the triangle. 2. **Formula:** Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin

1. **State the problem:** A goldfinch is flying 13 metres away from a bird box on top of a pole. The angle of depression from the goldfinch to the base of the pole is 32°. We need

1. **State the problem:** We need to find the length $g$ in a right triangle with a horizontal base of 2.8 cm, a left base angle of 25°, and a top right angle of 36°. The vertical

1. The problem is to solve for $a$ in the equation $\tan 9^\circ = \frac{a}{2.1}$. We want to find the value of $a$ to 2 decimal places. 2. The formula used here is the definition

1. **State the problem:** Calculate $2.6 \cos 12^\circ$ and round the result to 2 decimal places. 2. **Recall the formula:** The cosine function for an angle $\theta$ in degrees is

1. **State the problem:** Solve the inequality $$\frac{\sin x}{\tan x + 1} \geq 0$$ for $x$. 2. **Recall definitions and formulas:**

1. **State the problem:** Solve the inequality $$\frac{\sin x}{\tan x} + 1 > 0$$ for $x$. 2. **Recall the definitions and formulas:**

1. **State the problem:** We have two airplanes, A and B, both flying directly toward the same airport. Airplane A is 20 miles from the airport. The pilot of A sees B at a 45° angl

1. **State the problem:** Nicole shines a light from a lighthouse window 250 feet above water. Nick is on a ship 10 feet above water, and the angle of elevation to the light is 3°.

1. **State the problem:** We have a right triangle with a hypotenuse of length 40, an adjacent side to the marked angle of length 15, and we want to find the missing angle at verte