📏 trigonometry

1. **State the problem:** Solve the equation $$\cot 2x = - \sin x \cdot \cos x$$. 2. **Recall the formulas:**

1. Stating the problem: Solve the equation $$9 \cos(2\theta) + 3 = 5 \cos \theta$$ for $$\theta$$. 2. Recall the double-angle identity for cosine: $$\cos(2\theta) = 2\cos^2\theta -

1. **Stating the problem:** We are given several trigonometric identities and functions involving angle $\omega$ and variable $x$, and we need to analyze and solve the expressions

1. **State the problem:** We have a right triangle formed by the central pole (20 ft), the slant height (26 ft), and the base of the tent. We want to find the angle between the cen

1. **State the problem:** Calculate the size of angle BAC in each right-angled triangle using the tangent ratio. 2. **Recall the formula:** For a right triangle, \( \tan \theta = \

1. **Problem statement:** Find the trigonometric values of $\sin \omega$ and $\cos \omega$ given the point $M(-0.8, 0.6)$ on the unit circle in the second quadrant, where $\omega =

1. **State the problem:** Simplify the expression $$\frac{1+\tan^2(x)}{1+\cot^2(x)}$$. 2. **Recall the Pythagorean identities:**

1. مسئله: حل معادله $\tan x = 1 + \tan^2 x$ برای قسمت ب. 2. فرمول و قوانین مهم: می‌دانیم که $\tan^2 x + 1 = \sec^2 x$، پس معادله را می‌توان به صورت $\tan x = \sec^2 x$ نوشت.

1. **State the problem:** Calculate the value of $$\sin\left(-\frac{\pi}{12}\right) \cdot \csc\left(\frac{25\pi}{12}\right)$$. 2. **Recall definitions and properties:**

1. **State the problem:** Simplify the expression $\tan 40^\circ - \frac{\sin 40^\circ}{\cos 40^\circ}$. 2. **Recall the formulas:**

1. **State the problem:** We are given the function $y = -2 \cos(2x + 2\pi)$ and need to find its amplitude, period, and phase shift. 2. **Amplitude:** The amplitude of a cosine fu

1. **State the problem:** We have a flagpole on top of a building that is 9.0 m tall. From the top of the flagpole, the angle of depression to a spot on the ground is 49°. 2. From

1. **State the problem:** A construction worker 6.5 feet tall stands 8 feet away from a two-storey building 40 feet high. We need to find the angle of elevation from the worker's e

1. **State the problem:** We have a right triangle HIJ with a right angle at vertex I. Angle H is 27°.

1. **State the problem:** We need to find the angle $x$ in a right triangle with a right angle at $P$. The side opposite angle $x$ (OP) is 56, and the side adjacent to angle $x$ (P

1. **State the problem:** Simplify the expression $2\sin^2 x + \sin x - \cos^2 x$. 2. **Recall the Pythagorean identity:** $\sin^2 x + \cos^2 x = 1$.

1. **State the problem:** We need to graph a sine function with amplitude 4, period $\pi$, midline $y=-3$, and y-intercept at $(0,-3)$. The graph is not reflected over the x-axis.

1. **State the problem:** Simplify the expression $$\frac{\sin \theta}{1+\cos \theta} + \frac{1+\cos \theta}{\sin \theta} = 2 \csc \theta$$ and verify the equality. 2. **Recall imp

1. **State the problem:** We have a triangle split into two right triangles by a vertical height. The left triangle has an angle of 54° and a base of 8.2 cm. The right triangle has

1. The problem involves understanding and applying the Double Sine Law, which is used in solving triangles when two angles and one side or two sides and a non-included angle are kn

1. **State the problem:** Solve the equation $\tan x - \cot x = 0$ for $x=0$. 2. **Recall the definitions:**