Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 13

Four sketches of triangles are shown. The longest side of each triangle is 1 cm. The other length is given to 2 decimal places. a) Circle the value of sin 35° to 2 decimal places. b) Calculate the value of 𝑧, giving your answer to 1 decimal place.

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Video Transcript

Four sketches of triangles are shown. The longest side of each triangle is one centimeter. The other length is given to two decimal places. Part a) Circle the value of sin 35 degrees to two decimal places.

There is also a part b) to this question which we will look at later. The sine ratio or sin 𝜃 is equal to the opposite divided by the hypotenuse of a right-angled triangle. You might remember this from the SOH part of SOH CAH TOA. Of the four triangles shown, only two are right-angled. Therefore, we will focus on these two triangles. The longest side opposite the right angle is the hypotenuse. In both cases, this is equal to one centimeter.

The 0.57 centimeters in the first sketch is opposite the 35 degrees, whereas the 0.82 in the second sketch is next to or adjacent to the 35 degrees. As the second triangle has the adjacent and hypotenuse, this would enable us to calculate the cos or cosine of 35 degrees. Since we were asked to work out the sine of 35 degrees, we’ll use the first triangle.

Substituting in the values from this triangle gives us sin 35 is equal to 0.57 divided by one. Remember the opposite is the numerator and the hypotenuse is the denominator. Any number divided by one is equal to itself. Therefore, 0.57 divided by one is equal to 0.57. We can therefore see that sin 35 is equal to 0.57. The correct value correct to two decimal places is 0.57.

The second part of the question says the following. b) Calculate the value of 𝑧, giving your answer to one decimal place.

As the triangle is right angled, we can once again use SOH CAH TOA. Our first step is to label the sides of the triangle. The seven centimeters is the longest side and is opposite the right angle. Therefore, this is the hypotenuse. 𝑧 is the opposite as it is opposite the 35 degrees. Finally, the third side is the adjacent as it is adjacent or next to the 35-degree angle and the right angle.

In this question, once again, we want to use the sine ratio as we have the hypotenuse and want to calculate the opposite. As mentioned in part a), sin 𝜃 is equal to the opposite divided by the hypotenuse. Substituting in our values gives us sin 35 is equal to 𝑧 divided by seven. Multiplying both sides of this equation by seven gives us seven multiplied by sin 35 is equal to 𝑧. In part a) of the question we established that sin 35 was equal to 0.57 to two decimal places. This means that 𝑧 is equal to seven multiplied by 0.57.

There are lots of ways of working out this answer. One way would be using the traditional column method: 0.57 multiplied by seven. Seven multiplied by seven is equal to 49. So we put a nine in the hundreds column and carry the four. Seven multiplied by five is equal to 35. Adding the four gives us 39. We put a nine in the tenths column and carry the three. At this point, remember to include the decimal point in the correct place. Seven multiplied by zero is equal to zero. Adding three to this gives us three. Therefore, 0.57 multiplied by seven is equal to 3.99. We can therefore say that the length 𝑧 in the right-angled triangle is 3.99 centimeters.

An alternative method for multiplying seven by 0.57 would be to multiply 0.5 by seven and then 0.07 by seven. 0.5 multiplied by seven is equal to 3.5 or 3.50. If we think of this in money terms, seven lots of 50 pence is three pound 50. In the same way, 0.07 multiplied by seven is equal to 0.49. Once again, seven lots of seven pence is equal to 49 pence. Adding these two values 3.50 and 0.49 once again gives us 3.99.

In this question, we were asked to give our final answer to one decimal place. This means that we only want one number after the decimal point. The second nine is the deciding number. And as this is greater than five, we’ll round up. This means that the value to one decimal place is 4.0 or just four centimeters.

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