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Garret makes a ramp for his skateboard in the shape of a right triangle with a hypotenuse of 2ft, and a leg of 1f. He wants to use a trigonometric ratio to describe the relationship these two sides. Select all of the expressions that he could use. A. sin 30^(@) B. cos 45^(@) C. tan 30^(@) D. sin 45^(@) E. cos 60^(@) F. tan 45^(@)

Garret makes a ramp for his skateboard in the shape of a right triangle with a hypotenuse of 2ft 2 \mathrm{ft} , and a leg of 1f 1 \mathrm{f} . He wants to use a trigonometric ratio to describe the relationship these two sides. Select all of the expressions that he could use.\newlineA. sin⁡30∘ \sin 30^{\circ} \newlineB. cos⁡45∘ \cos 45^{\circ} \newlineC. tan⁡30∘ \tan 30^{\circ} \newlineD. sin⁡45∘ \sin 45^{\circ} \newlineE. cos⁡60∘ \cos 60^{\circ} \newlineF. tan⁡45∘ \tan 45^{\circ}

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Q. Garret makes a ramp for his skateboard in the shape of a right triangle with a hypotenuse of 2ft 2 \mathrm{ft} , and a leg of 1f 1 \mathrm{f} . He wants to use a trigonometric ratio to describe the relationship these two sides. Select all of the expressions that he could use.\newlineA. sin⁡30∘ \sin 30^{\circ} \newlineB. cos⁡45∘ \cos 45^{\circ} \newlineC. tan⁡30∘ \tan 30^{\circ} \newlineD. sin⁡45∘ \sin 45^{\circ} \newlineE. cos⁡60∘ \cos 60^{\circ} \newlineF. tan⁡45∘ \tan 45^{\circ}
  1. Find Triangle Angle: Determine the angle opposite the given leg.\newlineSince we have a right triangle with a hypotenuse of 2 ft2\,\text{ft} and a leg of 1 ft1\,\text{ft}, we can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c)(c) is equal to the sum of the squares of the other two sides (a(a and b)b): c2=a2+b2c^2 = a^2 + b^2.\newlineLet's assume the given leg of 1 ft1\,\text{ft} is side 'aa'. We need to find side 'bb'.\newlinec2=a2+b2c^2 = a^2 + b^2\newline1 ft1\,\text{ft}00\newline1 ft1\,\text{ft}11\newline1 ft1\,\text{ft}22\newline1 ft1\,\text{ft}33\newline1 ft1\,\text{ft}44\newlineNow we have the lengths of all sides: hypotenuse 1 ft1\,\text{ft}55, one leg 1 ft1\,\text{ft}66, and the other leg 1 ft1\,\text{ft}77.
  2. Calculate Triangle Angles: Calculate the angles of the triangle.\newlineWe can use the inverse trigonometric functions to find the angles. Since we have all the sides, we can use any of the trigonometric ratios. Let's find the angle opposite the given leg of 1 ft1\,\text{ft}.\newlinesin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\newlinesin⁡(θ)=12\sin(\theta) = \frac{1}{2}\newlineθ=sin⁡−1(12)\theta = \sin^{-1}(\frac{1}{2})\newlineθ=30 degrees\theta = 30\,\text{degrees} or π6\frac{\pi}{6} radians\newlineThe angle opposite the leg of 1 ft1\,\text{ft} is 30 degrees30\,\text{degrees}.
  3. Identify Trigonometric Expressions: Identify the correct trigonometric expressions.\newlineNow that we know the angle opposite the given leg is 3030 degrees, we can evaluate the given options:\newlineA. sin⁡30∘\sin 30^{\circ} - This is correct because sin⁡(30∘)=12\sin(30^{\circ}) = \frac{1}{2}, which is the ratio of the given leg to the hypotenuse.\newlineB. cos⁡45∘\cos 45^{\circ} - This is incorrect because the angle in the triangle is not 4545 degrees.\newlineC. tan⁡30∘\tan 30^{\circ} - This is correct because tan⁡(30∘)=13\tan(30^{\circ}) = \frac{1}{\sqrt{3}}, which is the ratio of the given leg to the other leg.\newlineD. sin⁡45∘\sin 45^{\circ} - This is incorrect because the angle in the triangle is not 4545 degrees.\newlineE. cos⁡60∘\cos 60^{\circ} - This is correct because sin⁡30∘\sin 30^{\circ}00, which is the ratio of the given leg to the hypotenuse, and sin⁡30∘\sin 30^{\circ}11 degrees is the complementary angle to 3030 degrees in the right triangle.\newlineF. sin⁡30∘\sin 30^{\circ}33 - This is incorrect and not applicable to the given triangle.

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