📏 trigonometry

1. **Problem 1:** Find the length of side $x$ in a right triangle with angles $39^\circ$ and $60^\circ$, side adjacent to $39^\circ$ is 38, side opposite $60^\circ$ is 5, hypotenus

1. **State the problem:** We have triangle EFG with angles $\angle G = 109^\circ$, $\angle F = 52^\circ$, and side $FG = 4$. We want to find side $g = EF$. 2. **Find the missing an

1. The problem asks to find the exact value of the trigonometric function for the angle $225^\circ$. 2. We recognize that $225^\circ$ is in the third quadrant where tangent is posi

1. **State the problem:** Solve the trigonometric equation $$7 \cos 2x - 24 \sin 2x = 12.5$$ for $$0 \leq x < 180^\circ$$, rounding answers to 1 decimal place. 2. **Use the formula

1. **State the problem:** Solve the equation $\cos(4x) = -1$ for $x$. 2. **Recall the cosine function properties:** The cosine function equals $-1$ at angles of the form $\pi + 2k\

1. **Problem 7:** A boat is due south of a lighthouse and sails on a bearing of 292° for 51 km until it is due west of the lighthouse. Find how far away it is from the lighthouse n

1. The problem is to determine the quadrant of an angle $t$ based on the signs of $\sin(t)$ and $\cos(t)$.\n\n2. Recall the unit circle quadrants and the signs of sine and cosine i

1. **Problem:** Find the value of $x$ in a right triangle where the side adjacent to the angle $48^\circ$ is 21 cm, and $x$ is the side opposite the $48^\circ$ angle. 2. **Formula:

1. **Problem statement:** Given $\sin A = \frac{6}{7}$ and $A$ is in quadrant II, find $\sin 2A$, $\cos 2A$, and $\tan 2A$. 2. **Recall formulas:**

1. **Stating the problem:** Simplify the expression $$\frac{3\sin 2x + 5\cos x}{5\sin 2x - 2\cos 2x}$$ and analyze the relationship with $$\tan x$$. 2. **Recall formulas:**

1. **State the problem:** Simplify the expression $$\frac{3\sin x + 6\cos x}{5\sin x - 2\cos x}$$ and evaluate given that $$\tan x = 7 - 1 = 6$$. 2. **Recall the formula and rules:

1. **State the problem:** Given $\sin(t) = \frac{3}{8}$, find (a) $\sin(-t)$ and (b) $\csc(-t)$. 2. **Recall important trigonometric identities:**

1. **Problem:** Evaluate the trigonometric function $\sin\left(-\frac{8\pi}{3}\right)$ using its period. 2. **Recall the period of sine:** The sine function has a period of $2\pi$,

1. **State the problem:** We have a lighthouse at point L, a boat at point A which is 589 feet from L, and another point B on the shoreline. The angles of elevation to the lighthou

1. **State the problem:** We have a lighthouse 111 feet tall and two points A and B on the water. From A, the angle of elevation to the top of the lighthouse is 8°, and from B (clo

1. **Problem statement:** We have a right triangle with vertices H, I, and J, where side HI is perpendicular to side IJ, forming a right angle at vertex I. Angle H measures 30 degr

1. **State the problem:** We need to find the distance $d$ from point $X$ on the shore to point $Y$ on the island. The triangle $ZXY$ has side $ZX = 285$ meters, angle $Z = 35^\cir

1. **Problem statement:** A plane travels 200 miles at a heading of 300° from an airport. We need to find how far west of the airport the plane is. 2. **Understanding headings:** H

1. **State the problem:** We have two triangles, EBR and BRO, sharing side BR = 24.58 km. We want to find the unknown side EB in triangle EBR using given angles and sides. 2. **Ide

1. **State the problem:** A canoeist needs to travel from the lake entry point to a campsite located 2.4 km north and 3.2 km west. We want to find the direction to head directly to

1. Problem: Given $\cos \theta = -\frac{15}{17}$ and $\theta$ is in Quadrant II, find $\sin(2\theta)$. 2. Formula: Use the double-angle identity for sine: