📏 trigonometry

1. **Problem statement:** Calculate the value of $\theta$ in the first triangle where the opposite side to $\theta$ is 13 and the hypotenuse is 19. 2. **Formula:** Use the sine rat

1. **Problem:** Calculate the value of $\theta$ to the nearest hundredth using trigonometric ratios and inverse trig functions for the first triangle with angle 45°, leg $x$, and h

1. **Problem statement:** The shadow of a skyscraper is 100 m longer when the angle of elevation of the sun is 40° than when it is 60°. We need to find the height of the skyscraper

1. The problem is to understand how to use the number 12.5 to get 34 degrees. 2. This likely involves a trigonometric function where 12.5 is related to an angle of 34 degrees.

1. **State the problem:** Calculate the measure of angle $\theta$ to the nearest degree using the given triangle and trigonometric relationships. 2. **Recall the tangent function:*

1. **Problem:** Prove the identity $\cot x (\cot x + \tan x) = \csc^2 x$. 2. **Recall the definitions and identities:**

1. **Problem statement:** A tree broken by the wind forms a right-angled triangle with the ground. The broken part of the tree makes an angle of 60° with the ground. The top of the

1. **Stating the problem:** Solve the equation $$6 \cot x \left(1 + \cot^2 x\right) = \frac{3\sqrt{3}}{2} \sin 2x$$ for $$0 \leq x \leq 2\pi$$. 2. **Recall identities:**

1. The problem is to solve the equation $$\sin\left(A + \frac{\pi}{2}\right) = \cos\left(\frac{A}{2}\right) + 2$$ for $$0 \leq A \leq 2\pi$$. 2. Recall the identity $$\sin\left(x +

1. **Problem:** Solve the equation $$12 \sin^2 x - 11 \sin x + 2 = 0$$ for $$\sin x$$. 2. **Formula and approach:** This is a quadratic equation in terms of $$\sin x$$. Let $$y = \

1. **Problem (i): Solve for $0 < \theta \leq 360^\circ$ the equation $4 \tan \theta + 5 \sin \theta = 0$.** 2. **Recall the definitions:**

1. The problem is to draw the function $y=\tan(x)$ in the interval $(-\pi, \pi)$.\n\n2. The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$. It has vertical asym

1. Énoncé : Résoudre dans $[\frac{\pi}{2}; \frac{3\pi}{2}]$ l'inéquation $\cos(x) < \frac{\sqrt{3}}{2}$.\n\n2. Rappel : $\cos(x)$ décroît de 0 à $\pi$ et remonte ensuite. La valeur

1. **State the problem:** We need to find the value of $d$ in the equation $y = a \sin(bx + c) + d$ that models the height $y$ of a student on a ferris wheel after $x$ seconds. 2.

1. **Problem:** In a right triangle, the opposite side to angle $\theta$ is 7 and the hypotenuse is 25. Find $\sin \theta$. 2. **Formula:** $\sin \theta = \frac{\text{opposite}}{\t

1. **State the problem:** Find the exact value of $$(\sec 30^\circ)(\cos 30^\circ)(\tan 60^\circ)(\cot 60^\circ)$$. 2. **Recall the definitions and values:**

1. **State the problem:** Find all $x$-intercepts of the function $$f(x) = 6 \sin^2 x + 3 \sin x - 3 = 0$$ on the interval $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$. 2. **Rewrite the equ

1. **State the problem:** We need to find the width of the river, which is the horizontal distance between the surveyor and the pole. 2. **Identify the known values:**

1. **State the problem:** Given $\sin(A+B) = 0.75$ and $\sin(A-B) = 0.43$, find the value of $\cos A \sin B$ to the nearest hundredth. 2. **Recall the sine addition and subtraction

1. The problem states that the angle is 81.4 degrees. 2. Since no further context or question is provided, we interpret this as identifying or using the angle value.

1. **State the problem:** Find the angle $\theta$ in a right triangle where the hypotenuse is 8 m and the adjacent side to $\theta$ is 5 m.