📏 trigonometry

1. **State the problem:** Simplify the expression $$\frac{2+\tan^2 x}{\sec^2 x} - 1 = g(x)$$. 2. **Recall the identity:** We know that $$\sec^2 x = 1 + \tan^2 x$$.

1. **State the problem:** We need to find the horizontal distance from the boat to the foot of the cliff. The cliff height is 55 m, and the angle of depression from the top of the

1. **Énoncé du problème :** Nous avons un triangle avec les côtés $A=5,9$, $B=3,4$ et l'angle $\alpha = 22^\circ$ opposé au côté $a$. Nous devons trouver l'angle $c$ (noté ici $\ga

1. **State the problem:** Evaluate the expression $\cos^2(10^\circ) + \cos^2(50^\circ) - \sin(40^\circ) \sin(80^\circ)$.\n\n2. **Recall relevant formulas:** Use the Pythagorean ide

1. **State the problem:** Given acute angles $A$ and $B$ such that $\sin(A-B)=0$ and $2\cos(A+B)-1=0$, find the values of $A$ and $B$. 2. **Use the given equations:**

1. The problem asks for the interval of the principal value of the function $\cos^{-1} x$, also known as the inverse cosine or arccosine function, and to draw its graph. 2. The pri

1. **Problem statement:** We have a field ABCD with given sides and angles. We need to find (a) the length of CD and (b) the angle ABD. 2. **Given data:**

1. The problem asks for the interval of the principal value of the function $\cos^{-1} x$ (arccosine of $x$) and to draw its graph. 2. The principal value of the inverse cosine fun

1. **State the problem:** We need to find the equation of a sinusoidal function based on the given graph description. 2. **Identify key features from the graph:**

1. The problem is to convert the expression $\frac{2\pi^c}{9}$ into degrees. 2. Recall that $\pi$ radians equals 180 degrees. To convert radians to degrees, multiply by $\frac{180}

1. **State the problem:** Convert the expression $\frac{2\pi^c}{9}$ into degrees. 2. **Recall the conversion formula:** To convert radians to degrees, use the formula:

1. **State the problem:** We have a ladder leaning against a wall forming a right-angled triangle. The ladder is the hypotenuse of length 5.2 m, and the base (distance from the wal

1. **State the problem:** We need to find the smallest possible angle $\theta$ between the plane's path and the ground, given the plane reaches a height of 10 km and flies a distan

1. **Problem Statement:** Determine whether to use the Law of Sines or Law of Cosines to find the indicated side length or angle measure in each triangle. 2. **Formulas:**

1. **Problem:** A construction worker looks up at the top of a crane 60 meters tall. The angle of elevation to the top is 35 degrees. Find the distance from the worker to the base

1. **State the problem:** Simplify the expression $$(\tanh x - 1)(\tanh x + 1) - 2 \tanh^2 x + \sec^2 x$$ using trigonometric identities. 2. **Recall relevant identities:**

1. **State the problem:** Simplify the expression $$\cos^2 m (\tan m - 1)(\tan m + 1) - 2 \tan^2 m + \sec^2 m$$. 2. **Recall formulas and identities:**

1. **Problem Statement:** Find the sine, cosine, and tangent of the given angles using the quadrant system. 2. **Formula and Rules:**

1. **Problem:** Given the diagram of △ABC with angles B = 50°, A = 80°, and side BC = 20 units, find the value of sec A. **Step 1:** Recall that \(\sec A = \frac{1}{\cos A}\).

1. **State the problem:** We want to analyze the function $$y = \sin(3x) \cos(2x)$$ and understand its behavior. 2. **Formula and identities:** We can use the product-to-sum identi

1. The problem is to find the value of $\cot 42^\circ$. 2. Recall that $\cot \theta = \frac{1}{\tan \theta}$, so we can find $\cot 42^\circ$ by calculating $\tan 42^\circ$ and then