📏 trigonometry

1. **State the problem:** Find the value of $\cot^{-1}(-\sqrt{3})$ in radians. 2. **Recall the definition:** The inverse cotangent function $\cot^{-1}(x)$ gives an angle $\theta$ s

1. **State the problem:** Find the exact values of the six trigonometric functions for a right triangle with sides 9, 40, and hypotenuse 41, angle $\theta$ opposite side 40. 2. **R

1. The problem is to find the values of $\sin \frac{5\pi}{8}$ and $\cos \frac{7\pi}{12}$ using half-angle formulas. 2. The half-angle formulas are:

1. **State the problem:** Solve the equation $$y = 3 \sin^2(x - \pi)$$ for $y$ in terms of $x$. 2. **Recall the formula and rules:** The function involves the square of the sine fu

1. **Problem Statement:** Sketch one full period of the function $y = 2 \sin(x) + 3$.

1. **Problem:** Find the coordinates of the point on the unit circle corresponding to an angle of $\frac{\pi}{3}$ radians. 2. **Formula and rules:** The unit circle is a circle wit

1. We start with the given formula: $$\cos(2t) = 1 - 2\sin^2(t)$$ 2. This is a double-angle identity for cosine, which relates the cosine of twice an angle to the sine squared of t

1. **Stel het probleem vast:** Los de vergelijking op $$\cos(2x) - 2 \sin^2(x) = 0$$. 2. **Gebruik trigonometrische identiteiten:** We weten dat $$\cos(2x) = \cos^2(x) - \sin^2(x)$

1. **State the problem:** Simplify the expression $5 \left( \csc \sqrt{x} \right)^3$. 2. **Recall the definition:** The cosecant function is $\csc \theta = \frac{1}{\sin \theta}$.

1. **Problem statement:** Dylan is 12 m away from a rock, and the angle of elevation to the top of the rock is 57°. We need to find the height of the rock. 2. **Formula used:** To

1. **State the problem:** We have a right triangle with sides 21, 20, and 29, where 29 is the hypotenuse. We need to find the ratio equivalent to $\sin(B)$. 2. **Recall the definit

1. **Problem statement:** Given right triangles with two sides, find $\sin$, $\cos$, and $\tan$ of angles $A$ and $B$ as common fractions. 2. **Recall:** In a right triangle, $c$ i

1. **State the problem:** We need to find the value of $\cos L$ in a right triangle $\triangle LMN$ where the right angle is at vertex $M$. The side opposite angle $L$ is $MN = 4$,

1. **Problem statement:** We need to express $\tan C$ as a fraction in simplest terms given a right triangle $CDE$ with right angle at $D$. The side opposite angle $C$ is $DE=12$,

1. **State the problem:** We need to find the length of side $x$ in a right triangle where the angle is $68^\circ$, the opposite side is $12.6$, and the adjacent side is $x$. The t

1. **Problem statement:** Given that $x$ is an acute angle and $\sin x = \frac{12}{13}$, find $\cos 2x$. 2. **Formula used:** The double-angle formula for cosine is

1. The problem is to write a trigonometric function that matches the given graph. 2. The general form of a sine function is $$f(x) = A \sin(B(x - C)) + D$$ where:

1. **State the problem:** We are given that $\sin \theta = \frac{\sqrt{2}}{2}$ and need to find $\tan \theta$. 2. **Recall the definitions and formulas:**

1. **State the problem:** We are given a right triangle with angle $x$ at vertex $B$, the side opposite $x$ is 7.4 units, and the side adjacent to $x$ is 5.6 units. We need to find

1. **State the problem:** We need to find a function of the form $f(x) = A \sec(Bx)$ that matches the given graph.

1. **State the problem:** We need to find a function of the form $f(x) = \csc(Bx - C)$ that matches the given graph. 2. **Identify key features from the graph:**