📏 trigonometry

1. **Problem statement:** We have a vertical pole DC on horizontal ground ABC, where ABC is a straight line. The angle of elevation of point D from A is 27°. We know the distances

1. **State the problem:** We have a right triangle with a hypotenuse of length 8 cm, a vertical side of length 5 cm, and we need to find the angle $z$ at the bottom-left corner bet

1. **State the problem:** We need to sketch the graph of the function $$f(x) = 2 \cos\left(x - \frac{\pi}{2}\right)$$ and understand its properties.

1. **State the problem:** Prove or simplify the identity:

1. **Problem (i): Solve $2 \sin \theta = 1$ for $\theta$.\n\n2. Divide both sides by 2: $$\cancel{2} \sin \theta = \frac{1}{\cancel{2}} \implies \sin \theta = \frac{1}{2}.$$\n\n3.

1. **State the problem:** We are given a right triangle with legs 32 and 60, hypotenuse 68, and we need to find $\tan B$ where $B$ is the angle opposite the leg of length 32. 2. **

1. **Problem a:** Prove that $\sin x \cos x \tan x = 1 - \cos^2 x$. 2. **Recall the identity:** $\tan x = \frac{\sin x}{\cos x}$.

1. **Problem:** Solve triangle $\triangle ABC$ given $\angle A = 38^\circ$, $\angle B = 90^\circ$, and side $c = 12.4$ cm. 2. **Step 1: Find $\angle C$**

1. **State the problem:** Find the exact value of $$\frac{\tan \frac{\pi}{10} + \tan \frac{\pi}{15}}{1 - \tan \frac{\pi}{10} \tan \frac{\pi}{15}}$$. 2. **Recall the formula:** This

1. **Problem statement:** We want to find the height of a mast on a hill. The hill's height is 542 m above sea level, and the observer is at 365 m above sea level. The angle of ele

1. **State the problem:** Verify the trigonometric identity: $$\frac{\sin\theta}{1-\cos\theta} + \frac{1-\cos\theta}{\sin\theta} = 2\csc\theta$$

1. **State the problem:** Calculate the value of the expression $$\sin(-\pi) - 7\sin^2\left(\frac{\pi}{4}\right)$$. 2. **Recall important formulas and values:**

1. **State the problem:** We need to evaluate the expression $$B = \frac{\sin\left(\frac{2\pi}{3}\right)}{\cos^2\left(-\frac{\pi}{3}\right)}$$ and simplify it. 2. **Recall importan

1. **State the problem:** Evaluate $\sin\sqrt{\frac{2\pi}{3}}$.\n\n2. **Recall the formula and rules:** The sine function takes an angle in radians and returns a value between $-1$

1. **State the problem:** Find the value of $\sec \frac{3\pi}{4}$. 2. **Recall the definition:** The secant function is the reciprocal of the cosine function:

1. **Stating the problem:** We have three right triangles with given sides and angles 30° and 60°, and we need to find missing sides using trigonometric ratios. 2. **Formulas and r

1. The problem is to understand the four quadrants of the coordinate plane and how positive and negative degrees relate to them. 2. The coordinate plane is divided into four quadra

1. The problem asks to find the value of $\sin 60^\circ$ as a fraction. 2. Recall the special angles in trigonometry: $\sin 60^\circ = \frac{\sqrt{3}}{2}$.

1. The problem asks to express $\sin 30^\circ$ as a fraction. 2. Recall the special angle values for sine: $\sin 30^\circ = \frac{1}{2}$.

1. **State the problem:** We are given a right triangle with vertices L, N, and M, where angle N is the right angle. The sides are LN = 5, NM = 12, and hypotenuse LM = 13. We need

1. **Express each angle in radian measure** The formula to convert degrees to radians is: