Question text Find trigonometric ratios of the angle [math: θ] heta in the right-angled triangle [math: △ABC] riangle ABC. (diagram is not to scale)[math: a=20,b=48,c=52]a={20}, b={48}, c={52} Note: Write your answers as a fraction [math: p]/[math: q]. [math: sin⁡(θ)=]\sin( heta)= [input] [math: 2052] \frac{20}{52} Correct answer. Well done. [math: cos⁡(θ)=]\cos( heta)= [input] [math: 4852] \frac{48}{52} Correct answer. Well done. [math: tan⁡(θ)=] an( heta)= [input] [math: 2048] \frac{20}{48} Correct answer. Well done.

Question

Question text Find trigonometric ratios of the angle [math: θ]	heta in the right-angled triangle [math: △ABC]	riangle ABC. (diagram is not to scale)[math: a=20,b=48,c=52]a={20}, b={48}, c={52} Note: Write your answers as a fraction [math: p]/[math: q]. [math: sin⁡(θ)=]\sin(	heta)= [input] [math: 2052] \frac{20}{52} Correct answer. Well done. [math: cos⁡(θ)=]\cos(	heta)= [input] [math: 4852] \frac{48}{52} Correct answer. Well done. [math: tan⁡(θ)=]	an(	heta)= [input] [math: 2048] \frac{20}{48} Correct answer. Well done.

BlackTom AI · Accepted Answer

The task involves finding the trigonometric ratios for the angle θ in the right-angled triangle △ABC with side lengths a = 20, b = 48, c = 52 where c is the hypotenuse. To reason through this, recall the standard definitions: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

First, consider sin(θ). Since the opposite side to θ is 20 and the hypotenuse is 52, sin(θ) = 20/52. This fraction reduces by dividing numerator and denominator by 4 to 5/13. So the simplified value is 5/13.

Next, for cos(θ). The adjacent side to θ is 48 and the hypotenuse is 52, so cos(θ) = 48/52. This reduces by dividing both by 4 to 12/13. Thus the simplified value is 12/13.

Then, for tan(θ). The tangent is the ratio of opposite to adjacent, so tan(θ) = 20/48. This reduces by dividing numerator and denominator by 4 to 5/12. Therefore the simplified value is 5/12.

If a student doesn’t simplify, they would write 20/52, 48/52, and 20/48 respectively, which are correct as fractions but not in simplest form. It’s common to simplify to smaller integers to present a cleaner exact value. A frequent pitfall would be mixing up opposite and adjacent sides or using the wrong hypotenuse; in this configuration, c = 52 is the hypotenuse, which makes the 20 and 48 assignments consistent with the standard definitions. Another misconception could be assuming all three ratios are simplified by the same factor, but each fraction has its own greatest common divisor (gcd) to reduce to simplest terms.

In summary, the properly simplified trigonometric ratios for θ are sin(θ) = 5/13, cos(θ) = 12/13, tan(θ) = 5/12. The unsimplified forms 20/52, 48/52, and 20/48 correspond to the same exact values before simplification, just not reduced to lowest terms.

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