📏 trigonometry

1. The problem is to simplify the expression $$\frac{\cos^2 a - \sin^2 a}{\cos^2 a (2 - \cos^2 a)}$$. 2. Recall the Pythagorean identity: $$\sin^2 a + \cos^2 a = 1$$.

1. **State the problem:** Simplify the expression $\sec(x) + \cos(x)$. 2. **Recall the definitions:** $\sec(x) = \frac{1}{\cos(x)}$.

1. **Problem statement:** Given the expression $6\cos 2x - 8\sin 2x$ which can be written as $R \cos(2x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$.

1. We are given a right triangle with vertices D (top-left), F (bottom-left, right angle), and E (bottom-right). Angles are labeled \(\alpha\) at D and \(\beta\) at E. 2. The sides

1. **Stel het probleem vast:** We moeten de ontbrekende goniometrische getallen invullen met behulp van de gegeven driehoek en daarna de lengte $x$ en hoek $\alpha$ berekenen. 2. *

1. **Stel het probleem vast:** We moeten de waarden van de goniometrische functies berekenen en hoeken bepalen volgens de gegeven opdrachten. 2. **Bereken tan 66°, cos 43°, sin 79°

1. **State the problem:** We have a right triangle with sides $a=5$, $b=12$, and hypotenuse $c$. We need to find $c$, $\sin(A)$, $\cos(A)$, and $\tan(A)$ where $A$ is the angle opp

1. **State the problem:** We have a right triangle with one leg $a=20$ and hypotenuse $c=29$. We need to find the other leg $b$ and the trigonometric ratios $\sin(B)$, $\cos(A)$, a

1. **State the problem:** We have a right triangle with sides $x=9$, $y=6$, and hypotenuse $z$. We need to find $\cos(\theta)$, $\sin(\theta)$, and $\tan(\theta)$ as exact fraction

1. **State the problem:** We have a right triangle with one angle of 36°, the side opposite this angle is 12, and the adjacent side to this angle is $x$. We need to find the length

1. **State the problem:** We have a right triangle P Q O with a right angle at P. Side QP is 73, side QO is 97, and we want to find the angle $x^\circ$ at vertex Q. 2. **Identify t

1. **State the problem:** We have a right triangle N-M-L with a right angle at M. Side N-M is 9.8, side N-L is 18, and we need to find the angle $x$ at vertex L. 2. **Identify the

1. **State the problem:** We have a right triangle $\triangle KLM$ with $\angle M = 90^\circ$, and side lengths $ML = 65$, $LK = 97$, and $KM = 72$. We need to find the sine of $\a

1. **State the problem:** We need to find the size of angle $x$ in a right-angled triangle where the vertical side is 7.9 cm and the base is 3.1 cm. 2. **Formula used:** To find an

1. **Problem:** Find the exact values of the six trigonometric functions for the angle $\frac{7\pi}{4}$ radians. 2. **Recall the six trigonometric functions:**

1. The problem asks which expressions have the same value as $\cos 50^\circ$.\n\n2. Recall the co-function identity: $\cos \theta = \sin (90^\circ - \theta)$. So, $\cos 50^\circ =

1. The problem asks which of the given trigonometric values equal 0. 2. Recall the basic trigonometric values for angles 0° and 90°:

1. **State the problem:** We need to find which of the given trigonometric expressions equal 1. 2. **Recall the values of basic trigonometric functions at special angles:**

1. **State the problem:** Given that $\tan \theta = \frac{3}{4}$ for an acute angle $\theta$, find integers $a$ and $b$ such that $\cos \theta = \frac{a}{b}$. 2. **Recall the defin

1. **State the problem:** We need to find the value of $\cos \theta$ for a right triangle where the adjacent side to angle $\theta$ is 5 and the hypotenuse is $\sqrt{29}$. 2. **Rec

1. **State the problem:** Given a triangle with sides $a=3$, $b=7$, and included angle $C=20^\circ$, find the remaining angle $A$, angle $B$, and side $c$. 2. **Formula used:** Use