📏 trigonometry

1. **Problem Statement:** We are given the expression $\frac{1 - \tan^2 \theta}{\sec^2 \theta} = \cos 2\theta$ and asked to:

1. Let's start by understanding what "non-permissible values" of an angle mean in math problems. 2. Non-permissible values are values of the angle that make the expression undefine

1. **Problem Statement:** Given the identity $$\frac{1 - \tan^2\theta}{\sec^2\theta} = \cos 2\theta,$$

1. **Problem Statement:** Given a right triangle JKL with right angle at J, sides JK = 39, JL = 52, and hypotenuse KL = 65, express the trigonometric ratios $\cos K$ and $\sin L$ a

1. **State the problem:** We are given a right triangle with vertices J, K, and L, right angle at J, and side lengths JK = 39, JL = 52, KL = 65. We know \(\cos K = \frac{3}{5}\) an

1. **Problem Statement:** Express the cosine of angle $K$ in simplest fractional form given a right triangle with sides opposite vertices $J$, $K$, and $L$ measuring 65, 52, and 39

1. **State the problem:** We have a right triangle SRQ with a right angle at R. Side SR = 35, angle Q = 40°, and we need to find side SQ = x.

1. **State the problem:** We have a point $P(k, 5)$ in the second quadrant such that the distance from the origin $O$ to $P$ is 13 units. We need to find the value of $k$.

1. **Stating the problem:** We have a point $P(k, 5)$ in the second quadrant such that the distance from the origin $O$ to $P$ is 13 units. We need to find the value of $k$.

1. **State the problem:** We are given an angle $\theta$ with terminal point $\left(\frac{3}{6}, \frac{\sqrt{11}}{6}\right)$ on the unit circle. We need to find which value is equi

1. **Problem:** Given a right triangle ABC with a right angle at A, AB = 3 cm, and BC = 6 cm. Find $\sin B$. 2. **Formula:** In a right triangle, $\sin$ of an angle is the ratio of

1. **Problem statement:** We have a right triangle ABC with a right angle at C, angle B = 67°, side b = 10.6 opposite to angle C, and side a opposite to angle A. We need to find si

1. **Problem statement:** A telephone pole is supported by two wires forming a 70° angle at the top. The ends of the wires on the ground are 20 m apart. One wire makes a 30° angle

1. **State the problem:** We have a right triangle with angles 70° and 30°, and the hypotenuse length is 5. We need to find the length of side $x$ adjacent to the 30° angle. 2. **R

1. **Problem statement:** Solve the equation $$4 \cos x - 2 \cos 2x = 0$$ for $$0 \leq x \leq \pi$$. 2. **Formula and identities:** Recall the double-angle identity for cosine:

1. Given: $\sin \alpha = -\frac{1}{3}$ and $\cos \beta = -\frac{1}{2}$. We need to find $\sin(\alpha + \beta) \sin(\alpha - \beta)$. 2. Use the formulas:

1. **State the problem:** We are given the function $y = \tan 5x$ and asked to analyze or solve it. 2. **Recall the formula and properties:** The tangent function is defined as $\t

1. The problem is to find the value of $\sin x$ given some condition, but the condition is incomplete in the question. 2. The sine function, $\sin x$, represents the ratio of the o

1. **State the problem:** Solve the trigonometric equation $$\cos \alpha - 4 \sin^2 \alpha = 3$$ for $$0^\circ \leq \alpha \leq 360^\circ$$. 2. **Use the Pythagorean identity:** Re

1. Planteamos el problema: Resolver la ecuación $\tan 2x = \cot 2x$ en el intervalo $[0^\circ, 360^\circ]$. 2. Recordemos que $\cot \theta = \frac{1}{\tan \theta}$, por lo que la e

1. **State the problem:** We need to find all values of $x$ in the interval $0 \leq x \leq 2\pi$ such that $$\cos\frac{11\pi}{31} = \sin x.$$\n\n2. **Use the identity:** Recall tha