📏 trigonometry

1. **State the problem:** Calculate $\tan 150^\circ - \sin 135^\circ$.\n\n2. **Recall the formulas and values:**\n- $\tan(180^\circ - \theta) = -\tan \theta$.\n- $\sin(180^\circ -

1. **State the problem:** We are given a right triangle with vertices A, B, and C, where angle C is the right angle.

1. **Problem statement:** We have two right triangles. For the first triangle ABC with right angle at C, sides BC=8, AC=6, and hypotenuse AB=10, find the six trigonometric ratios f

1. **State the problem:** We have a truck on a ramp that forms a right triangle. The ramp length (hypotenuse) is 75 feet, and the vertical height is $x$. We want to find the tippin

1. **State the problem:** We need to find the length of the wire that supports the tower. The wire is the hypotenuse of a right triangle where the side adjacent to the 60° angle is

1. **Problem statement:** Find all angles that satisfy the given condition (the problem is incomplete, so we assume a general approach to finding all angles in trigonometry. 2. **G

1. **Problem Statement:** From a window 16 ft above the ground, a student measures the angles of elevation and depression to the top and base of a nearby tree. We need to find: a)

1. **State the problem:** We need to find how much higher the longer brace reaches compared to the shorter brace. 2. **Recall given data:**

1. **Problem statement:** We have two braces placed against the side of a house forming angles of 45° and 30° with the vertical wall. The shorter brace is 8 m long at a 45° angle.

1. **State the problem:** We need to find the angle $\theta$ in a right triangle where the legs adjacent to $\theta$ are both 3 units long. 2. **Identify the sides relative to $\th

1. **Problem Statement:** Find the value of angle $\theta$ in a right triangle where the side opposite to $\theta$ is 11.9 and the side adjacent to $\theta$ is 10.

1. Énoncé du problème : On a \( \csc x = \frac{2\sqrt{3}}{3} \) avec \( x \in [0, \frac{\pi}{2}] \). On doit trouver : a. \( \cos x \)

1. Énoncé du problème : On a \( \csc x = \frac{2\sqrt{3}}{3} \) avec \( x \in [0, \frac{\pi}{2}] \). Trouver : a. \( \cos x \)

1. The problem is to analyze and graph the function $f(x) = 2 \cos(5x - 1)$.\n\n2. The general form of a cosine function is $y = A \cos(\beta (x - c)) + D$, where:\n- $A$ is the am

1. **State the problem:** Solve the equation $$\tan^2(x) - \sin^2(x) = \sin^4(x) \sec^2(x)$$ for $x$. 2. **Recall definitions and identities:**

1. **State the problem:** Prove that $\tan^2(x) - \sin^2(x) = \sin^4(x) \sec^2(x)$.\n\n2. **Recall definitions and identities:** - $\tan(x) = \frac{\sin(x)}{\cos(x)}$

1. **State the problem:** Find the angle of elevation $x$ of the sun when a tree 10 yards tall casts a shadow 14 yards long. 2. **Identify the triangle sides:** The tree height is

1. **Problem statement:** A surveyor moves 140 feet away from the base of a pole and measures the angle of elevation to the top of the pole as 44° using a transit 4 feet tall. Find

1. **State the problem:** Find all solutions of the equation $$2\sin^2 y = 2 + \cos y$$ on the interval $$[0, 2\pi)$$. 2. **Rewrite the equation using the Pythagorean identity:** R

1. **Problem 4: Find the angle of elevation from the device to the top of the tree.** Given:

1. Solve the proportion $\frac{3a+8}{6} = \frac{a-7}{9}$. Use cross multiplication: $9(3a+8) = 6(a-7)$.