📏 trigonometry

1. **State the problem:** Find the length of side $x$ in a right triangle where the hypotenuse is 13 cm and the angle adjacent to $x$ is 54° using the cosine ratio.

1. **State the problem:** A surveyor needs to find the distance between points A and B, but a house obstructs the direct path. Given are side lengths $a=58$ feet, $b=75$ feet, and

1. The problem asks to find the exact value of the sine ratio for the angle of 1.5 radians in standard position. 2. Recall that the sine of an angle in standard position on the uni

1. The problem asks to sketch an angle in standard position whose terminal arm passes through the point (1,5). 2. An angle in standard position has its vertex at the origin (0,0) a

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