📏 trigonometry

1. **State the problem:** Jack is on top of a 65 m high cliff. He observes a man swimming out to sea at an angle of depression of 51° and a boat at an angle of depression of 30°. W

1. **State the problem:** Evaluate $\tan\left(-\frac{\pi}{6}\right)$ without using a calculator. 2. **Recall the formula and properties:** The tangent function is odd, meaning $\ta

1. The problem asks to evaluate $\sin\left(\frac{5\pi}{6}\right)$ without using a calculator. 2. Recall the sine function properties and the unit circle: $\sin(\theta)$ is positive

1. **State the problem:** We need to find $\sin(x^\circ)$ for a right triangle where the opposite side to angle $x$ is 7, the adjacent side is 24, and the hypotenuse is 25. 2. **Re

1. **State the problem:** Prove or verify the identity $$ (1 + \tan^2 \theta) \cos^2 \theta = 1 $$. 2. **Recall the Pythagorean identity:** We know that $$ 1 + \tan^2 \theta = \sec

1. **State the problem:** We want to find how high the drawbridge rises when the angle $x$ is $30^\circ$, $45^\circ$, and $60^\circ$. The drawbridge half is the hypotenuse of a rig

1. **State the problem:** We are given a triangle with sides 8 units and 7 units enclosing an angle $\theta$. The area of the triangle is $14\sqrt{3}$ square units. We need to find

1. The problem is to understand when to use sine, cosine, or tangent in a right triangle. 2. The key is to remember the definitions of sine, cosine, and tangent based on the angle

1. **State the problem:** Given $\sin \theta = -\frac{3}{5}$ and $\cos \theta = \frac{4}{5}$, find $\tan \theta$ for $0^\circ \leq \theta \leq 360^\circ$ without using a calculator

1. The problem asks to rotate the angle $-526^\circ$ to an equivalent angle within $10^\circ$. 2. To find an equivalent angle within a standard range, we add or subtract full rotat

1. **State the problem:** A boat starts at point A, 994 feet from the lighthouse base L. The angle of elevation to the lighthouse beacon from A is 12°. Later, at point B (between A

1. **State the problem:** Myesha stands 13 meters from a building. The angle of elevation to the roof (point A) is 29°, and to the top of the antenna (point B) is 43°. Her eye heig

1. **State the problem:** We need to calculate the length $b$ in the given triangle, where one side is 23 mm and angles are given as 34°, 48°, 97°, and 102°. 2. **Analyze the trian

1. **State the problem:** We have a right triangle with \(\angle C = 90^\circ\), \(\angle A = 30^\circ\), and \(\angle B = 60^\circ\). The side opposite \(\angle B\) (side AC) is 1

1. **Problem Statement:** We have a right triangle ABC with a right angle at C.

1. **State the problem:** We are given a right triangle with angle $\angle A = 30^\circ$, side $AC = 18$ units (adjacent to $\angle A$), and we need to find the measure of $\angle

1. **State the problem:** We are given the equation $g \sin 20^\circ = 7$ and need to find the value of $g$ to 2 decimal places. 2. **Formula and rules:** To isolate $g$, divide bo

1. The problem is to calculate the value of $$\frac{20}{\cos 70^\circ}$$ rounded to 1 decimal place. 2. Recall that $$\cos 70^\circ$$ is the cosine of 70 degrees. We use the cosine

1. **State the problem:** We have a right triangle PQR with \(\angle R = 90^\circ\), \(\angle P = 24^\circ\), and hypotenuse \(PQ = 7.5\) cm. We need to find the lengths of sides \

1. **State the problem:** Find the value of $\tan \theta$ given the point $(-\sqrt{33}, -4)$ on the terminal side of angle $\theta$. 2. **Recall the formula:** The tangent of an an

1. The problem is to solve the equation $$6 \arcsin x = \pi$$ for $x$. 2. Recall that $\arcsin x$ is the inverse sine function, which returns an angle $\theta$ such that $\sin \the