📏 trigonometry

1. The problem asks for the expression for the length \(AC\) in triangle \(ABC\).\n\n2. Given: \(\triangle ABC\) is a right triangle with right angle at \(B\), \(\angle BAC = 50^\c

1. The problem asks us to find the exact value of \(\tan\left(\dfrac{7\pi}{12}\right)\) using an angle addition or subtraction formula.\n\n2. We express \(\dfrac{7\pi}{12}\) as a s

1. **State the problem:** Solve the equation $$-4\cos(5x) + 1 = 1$$ for all solutions in radians, where $n$ is any integer. 2. Simplify the equation:

1. **State the problem:** We need to find the distance from the UFO to satellite KA-12 given the triangle with vertices UFO, KA-12, and SAL-1. 2. **Given information:**

1. The problem is to verify the six trigonometric ratios (sin θ, cos θ, tan θ, csc θ, sec θ, cot θ) for two right triangles given the side lengths. ### First Triangle (baseball bal

1. **Problem 1:** Find all six trigonometric ratios for a right triangle where the opposite side is 21 m, adjacent side is 28 m, and hypotenuse is 35 m. 2. Given values are:

1. **State the problem:** We have two right triangles and need to find all six trigonometric ratios (sin, cos, tan, csc, sec, cot) for each angle $\theta$ given the sides.

1. **State the problem:** Given a right triangle with an adjacent leg of length 7 cm and a hypotenuse of length 13 cm, find all six trigonometric ratios for angle $\theta$. 2. **Fi

1. The problem states that $\sin\theta = \sin\alpha$. 2. Using the sine function properties, if $\sin A = \sin B$, then $A = B + 2k\pi$ or $A = \pi - B + 2k\pi$, where $k$ is any i

1. The problem is to determine the reference angle for $\theta = \frac{11\pi}{6}$ and find $\sin \theta$, $\cos \theta$, and $\tan \theta$.\n\n2. First, find the reference angle. R

1. Stating the problem: Find the reference angle for $\theta = \frac{11\pi}{6}$, and compute $\sin \theta$, $\cos \theta$, and $\tan \theta$. 2. Reference angle: The reference angl

1. The problem is to understand how to apply reciprocal identities in trigonometry. 2. Reciprocal identities relate trigonometric functions to their reciprocals:

1. Let's start by defining what reciprocal identities are. 2. Reciprocal identities are fundamental trigonometric identities that relate the basic trigonometric functions to their

1. **State the problem:** Simplify the expression $$\sec \Theta + \frac{\csc \Theta \tan \Theta}{\sec^2 \Theta}$$ and verify if it equals $$\cot \Theta \csc \Theta$$. 2. **Rewrite

1. Problem statement: Given a smoke stack centered at the origin with four radial labels C-4 (top-left), C-3 (top-right), C-2 (bottom-right) and C-1 (bottom-left) and radial lines

1. **State the problem:** Prove the identity $$\sec \Theta + \csc \Theta \tan \Theta = \cot \Theta \csc \Theta.$$\n\n2. **Write each trigonometric function in terms of sine and cos

1. Problem: Prove that $\frac{\sec \theta + \csc \theta \tan \theta}{\sec^2 \theta} = \cot \theta \csc \theta$. 2. Rewrite the left-hand side in terms of sine and cosine to simplif

18(a) Show that \(\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta\). 1. Start with the triple-angle formula for cosine:

1. **Show that** $\sin 3x \equiv 3 \sin x - 4 \sin^3 x$. Start from the triple angle identity for sine:

1. Let's start by understanding the sine function $\sin x$. It represents the ratio of the opposite side to the hypotenuse in a right triangle for an angle $x$. 2. The sine of an a

1. Dada a equação $$4\cos^2(x)\sin(x) - \sin(x) = 0$$, queremos encontrar os valores de $x$ que satisfazem essa igualdade. 2. Fatore $\sin(x)$ em evidência na expressão: