Video: Using Trigonometric Ratios to Find the Length of the Side Opposite the Angle

Find the length of 𝐡𝐢 giving the answer to two decimal places.

02:11

Video Transcript

Find the length of 𝐡𝐢 giving the answer to two decimal places.

As the triangle is right angled, we can use the trigonometrical ratios: sine, cosine, and tangent, which are shortened on the calculator to sin, cos, and tan. In any right-angled triangle, sin πœƒ is equal to the opposite divided by the hypotenuse, cos πœƒ is equal to the adjacent divided by the hypotenuse, and tan πœƒ is equal to the opposite divided by the adjacent.

If we begin by labelling the side 𝐡𝐢 as the letter π‘₯, we can now look at the three sides and decide which one is the opposite, which one is the adjacent, and which one is the hypotenuse. The hypotenuse is the longest side and is opposite the right angle. In this case, 𝐴𝐢 is the hypotenuse. As side 𝐡𝐢 is opposite the angle 47 degrees, that is the opposite side. And finally, side 𝐴𝐡 is the adjacent.

In this question, we know the length of 𝐴𝐢, 15 centimetres. And we are trying to work out the length of 𝐡𝐢, π‘₯ centimetres. Therefore, we will use the opposite and the hypotenuse, sin πœƒ. Substituting in the values into sin πœƒ equals the opposite over hypotenuse gives us sin 47 equals π‘₯ divided by 15. Multiplying both sides by 15 gives us π‘₯ equals 15 multiplied by sin 47.

Ensuring our calculator is in degree mode, typing in 15 multiplied by sin 47 gives us an answer of π‘₯ equals 10.97. Therefore, the length of 𝐡𝐢 in the triangle is 10.97 centimetres to two decimal places. These trigonometrical ratios can be used to find missing angles and missing lengths in any right-angled triangle.

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