📏 trigonometry

1. **State the problem:** We need to find the values of the cotangent function for the angles 0°, 30°, 45°, 60°, and 90°. 2. **Recall the definition:** The cotangent of an angle $\

1. **Problem Statement:** We have two figures with an inclined pole. In Figure 1, the pole makes an angle $x$ with the ground and the height of the top of the pole from the ground

1. **State the problem:** We are given two sinusoidal functions modeling physical phenomena: the length of day $L(d)$ on planet Kepple as a function of day $d$, and the volume of h

1. **State the problem:** We are given the function $H(t) = -4\sin\left(\frac{\pi}{3}(t - 1)\right) + 4$ describing Chloe's height in a wave pool over time $t$ seconds. 2. **Find t

1. **Problem statement:** We are given a sinusoidal function representing height over time and asked to create the equation for height, find the radius of the wheel, initial height

1. **Problem statement:** We have two tracking stations A and B, 49 miles apart. A satellite is above the ground at point C. The angles of elevation from A and B to the satellite a

1. **Problem statement:** We are given bearings and travel times between three cities A, B, and C. The bearing from A to B is S 65° E, and from B to C is N 50° E. A car travels fro

1. **State the problem:** Solve the trigonometric equation $$3\cos x + 2\sin 2x = 1$$ for $x$. 2. **Recall the double-angle identity:** $$\sin 2x = 2\sin x \cos x$$.

1. **Problem statement:** Given $\cot(\theta - 30^\circ) = \frac{1}{\sqrt{3}}$, find $\sin \theta$. 2. **Recall the cotangent values:**

1. The problem is to find the period of the function $\sin 2x$. 2. Recall that the general form of the sine function is $\sin bx$, where $b$ affects the period.

1. The problem involves analyzing the fish catch results of three boats represented by the function $f(x) = a \sin kx$ where $x$ is the direction in degrees and $f(x)$ is the catch

1. **Problem statement:** Prove that $\arcsin x + \arccos x = 90^\circ$ or equivalently $\frac{\pi}{2}$ radians. 2. **Recall definitions:**

1. **Problem statement:** Anna and Julia start at point P. Point Q is directly north of P. Anna walks first on a bearing of 330°, then changes to 040° to reach Q. Julia walks 3 km

1. **State the problem:** Solve the trigonometric equation $$3 \sin^2 x - 2 = \sin x$$ for $$0^\circ \leq x < 360^\circ$$. 2. **Rewrite the equation:** Let $$s = \sin x$$. The equa

1. The problem involves understanding the trigonometric relationships in a right triangle with angle $\alpha$ at vertex A, opposite side $X$, adjacent side $Y$, and hypotenuse $R$.

1. **Problem Statement:** Find the value of $\sin(\pi - \theta)$ in terms of $\sin \theta$ or $\cos \theta$. 2. **Formula and Explanation:** Use the sine subtraction formula or the

1. **Problem Statement:** Given that $\sin \theta = \frac{3}{5}$ and $\theta$ is in the first quadrant, find $\cos \theta$. 2. **Formula and Rules:** We use the Pythagorean identit

1. The problem asks for the value of $\tan 45^\circ$.\n\n2. Recall the definition of tangent in a right triangle: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.\n\n3. For

1. The problem asks for the value of $\cos 60^\circ$.\n\n2. Recall that cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.\n\n3. The cosine

1. The problem asks for the value of $\sin 30^\circ$. 2. Recall the sine function in trigonometry gives the ratio of the length of the opposite side to the hypotenuse in a right tr

1. The problem is to convert the angle $\frac{2\pi}{3}$ radians into degrees. 2. The formula to convert radians to degrees is: