📏 trigonometry

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:

1. مسئله (الف): اگر $\sin \alpha = \frac{3}{5}$ و زاویه $\alpha$ در ربع دوم باشد، مقدار $\cos \alpha$ و $\tan \alpha$ را بیابید. 2. فرمول‌های مورد استفاده:

1. **State the problem:** Solve the equation $$2(\sin x)^2 - 5 \cos x + 1 = 0$$ for $$x$$ in the interval $$[0, 2\pi]$$, expressing answers as multiples of $$\pi$$. 2. **Use the Py

1. **State the problem:** Solve the equation $$2 \sin t \cos t + \sin t - 2 \cos t - 1 = 0$$ for $$t$$ in the interval $$[0, 2\pi]$$, where the solutions must be multiples of $$\pi

1. **State the problem:** Jack and Jill are on opposite sides of an 88 metre deep canyon. Jack sees the trail guide at an angle of depression of 45° and Jill sees the trail guide a