📏 trigonometry

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

1. **State the problem:** Solve the equation $\cos \theta + 1 = 0$ for all values of $\theta$ where $\theta$ is in radians. 2. **Rewrite the equation:**

1. The problem is to identify the correct Pythagorean identities from the given options. 2. The fundamental Pythagorean identity is:

1. Problem statement: P is 12 km due north of Q, the bearing of R from P is 135^\circ and from Q is 120^\circ; find (i) angle PRQ and (ii) distance PR. 2. Set a coordinate frame: p

1. The problem is to sketch the graph of the function $y = \cos x$ for $0 \leq x \leq 360^\circ$.\n\n2. The cosine function is periodic with period $360^\circ$, and it oscillates b