📏 trigonometry

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:**

1. **State the problem:** We need to evaluate the function $f(x) = 1 + \tan(2x)$ at $x = \pi$. 2. **Recall the formula:** The function is given by

1. **State the problem:** You are in a hot air balloon 600 feet above the ground, looking down at a friend who is 1000 feet horizontally from the point directly under the balloon.

1. **Problem statement:** (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ can be rewritten as $$3 \sin^2 \theta + 8 \sin \theta - 3 = 0$$.

1. **State the problem:** Show that the equation $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$ can be written as $$5 \cos^2 \theta - \cos \theta - 4 = 0$$.

1. **Problem (i): Solve for $-\frac{\pi}{2} < x < \frac{\pi}{2}$ the equation $\tan^2(2x + \frac{\pi}{4}) = 3$.** 2. The formula used is $\tan^2 \theta = k$ which implies $\tan \th

1. **Problem Statement:** Find the values of the trigonometric functions sin \(\theta\), cos \(\theta\), tan \(\theta\), sec \(\theta\), csc \(\theta\), and cot \(\theta\) for the

1. **State the problem:** We need to find the measure of the angle labeled with a question mark (?) in a right triangle where the opposite side length is 70 and the adjacent side l

1. **State the problem:** Solve for $x$ in a right triangle where the hypotenuse is 13, one angle is 30°, and the right angle is 90°.

1. **State the problem:** Simplify and verify the equation $$\frac{\sin(x) \cdot \cos(x)}{\cos(2x) + \sin(x^2)} \cdot \frac{\sin(x) + \sin(x^2)}{\frac{1}{2} \sin(2x) + \cos(x)} + 1

1. The problem is to verify the trigonometric identity: $\cos 2x = 1 - 2 \sin^2 x$. 2. The double-angle formula for cosine states that:

1. **הבעיה:** במשולש ישר זווית ABD נתון: $\angle A=90^\circ$, $\angle B=68^\circ$, $BD=7$ ס"מ. 2. **נמצא את אורך הצלע AD:**

1. **Problem:** Find $\sin 30^\circ + \cos 45^\circ$ without using a calculator. 2. **Recall the exact values:**

1. Problem 1: An airplane is approaching a runway with an angle of depression of 3° and altitude 500 feet. We want to find the straight line distance to the runway. 2. Use the righ

1. **State the problem:** Solve the equation $$2 \sin(2x) = 3 \tan(x)$$ for $$x$$ in the interval $$[0, \pi]$$. 2. **Recall formulas and identities:**

1. **State the problem:** Find the value of $A$ such that the equation $\cos^2 x (1 + \cot^2 x) = A$ is an identity.

1. **State the problem:** Verify the identity $$\frac{\sec^2 \theta - 1}{\sec^2 \theta} = ?$$ 2. **Recall the Pythagorean identity:** $$\tan^2 \theta + 1 = \sec^2 \theta$$

1. **State the problem:** Solve the equation $$\cos 2\theta + \cos \theta + 1 = 0$$ for all values of $$\theta$$ in radians. 2. **Use the double-angle formula:** Recall that $$\cos