Question

4/13
categorize the following images by what you would use to solve for x.

options (vertical buttons):
sin or sin⁻¹
cos or cos⁻¹
tan or tan⁻¹
pythagorean theorem

images (triangles with x, right angles, sides/angles labeled):

1. triangle: leg 38, leg 42, hypotenuse x (right angle)
2. triangle: leg 11, leg x, angle 48° (right angle)
3. triangle: angle 59°, leg 12.5, hypotenuse x (right angle)
4. triangle: angle 59°, hypotenuse 12, leg x (right angle)
5. (partially visible) triangle with x, right angle

Explanation:

Step1: Analyze the first triangle (38, 42, x, right - angled)

This is a right - angled triangle with two legs of lengths 38 and 42, and we need to find the hypotenuse \(x\). The Pythagorean theorem states that for a right - angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). So we use the Pythagorean theorem here. Let \(a = 38\), \(b = 42\), then \(x^{2}=38^{2}+42^{2}=1444 + 1764=3208\), and \(x=\sqrt{3208}\approx56.64\).

Step2: Analyze the second triangle (11, x, 48° right - angled)

In a right - angled triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 48^{\circ}\), the opposite side to the angle is 11, and the adjacent side is \(x\). So \(\tan(48^{\circ})=\frac{11}{x}\), which can be rewritten as \(x=\frac{11}{\tan(48^{\circ})}\). So we use \(\tan\) or \(\tan^{-1}\) (if we were solving for the angle, but here we solve for \(x\) using \(\tan\)).

Step3: Analyze the third triangle (21.5, 59°, x, right - angled)

We know the adjacent side to the 59° angle is 21.5, and we need to find the hypotenuse \(x\). The cosine of an angle \(\theta\) in a right - angled triangle is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). So \(\cos(59^{\circ})=\frac{21.5}{x}\), and \(x = \frac{21.5}{\cos(59^{\circ})}\). So we use \(\cos\) or \(\cos^{-1}\).

Step4: Analyze the fourth triangle (12, 59°, x, right - angled)

We know the hypotenuse is 12, and we need to find the adjacent side \(x\) to the 59° angle. Using \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\cos(59^{\circ})=\frac{x}{12}\), so \(x = 12\cos(59^{\circ})\). So we use \(\cos\) or \(\cos^{-1}\).

Step5: Analyze the fifth triangle (let's assume the given side is opposite to the angle, and we need to find \(x\) (hypotenuse))

If we know the opposite side to an angle and we want to find the hypotenuse, we use \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So we use \(\sin\) or \(\sin^{-1}\).

Answer:

• First triangle (38, 42, x): Pythagorean Theorem
• Second triangle (11, x, 48°): \(\tan\) or \(\tan^{-1}\)
• Third triangle (21.5, 59°, x): \(\cos\) or \(\cos^{-1}\)
• Fourth triangle (12, 59°, x): \(\cos\) or \(\cos^{-1}\)
• Fifth triangle (opposite side, angle, x): \(\sin\) or \(\sin^{-1}\)