Question

categorize the following images by what you would use to solve for x. options (7) sin or sin⁻¹ cos or cos⁻¹ tan or tan⁻¹ pythagorean theorem

Explanation:

Step1: Analyze the bottom triangle

The bottom triangle is a right triangle with hypotenuse \( c = 25 \) and one leg \( a = 14 \). We need to find the other leg \( x \) (let's call it \( b \)). The Pythagorean Theorem states that \( a^{2}+b^{2}=c^{2} \), so we can use this to solve for \( x \) (since we know two sides of a right triangle, the Pythagorean Theorem is applicable here).

Step2: Check other methods

• For \( \sin \) or \( \sin^{-1} \), we need an angle and a side or two sides related to an angle (opposite and hypotenuse or opposite and adjacent). Here we have two sides (leg and hypotenuse) of a right triangle, not related to an angle in a way that sine would be the first choice.
• For \( \cos \) or \( \cos^{-1} \), we need adjacent and hypotenuse or adjacent and opposite related to an angle. Again, with two sides (leg and hypotenuse) of a right triangle, cosine is not the first choice.
• For \( \tan \) or \( \tan^{-1} \), we need opposite and adjacent sides related to an angle. We don't have an angle here, so tangent is not applicable.
• Since we have a right triangle with two known sides (one leg and hypotenuse), the Pythagorean Theorem (\( x=\sqrt{c^{2}-a^{2}}=\sqrt{25^{2}-14^{2}} \)) is the correct method to solve for \( x \).

Answer:

The bottom triangle (the one with sides 14, 25, and right angle) should be categorized under "Pythagorean Theorem"