Hi, I need help with figuring out how I calculated this question because my math teacher is asking "But my question is how do you get √2/2
How did you come up with that calculation?"
Here's the question and my work for it:
Chang is participating in a charity bicycle road race. The route starts at Centreville and travels east for 13 km to Eastdale. He then makes a 135° turn and heads northwest for another 18 km, arriving at Northcote. The final leg of the race returns to Centreville.
a) What is the total length of the race, to the nearest tenth of a kilometre? (2 marks)
b) What are the angles in the triangle formed by the three towns, to the nearest degree? (2 marks)
a)
CN2=132+182-2-13-18-cos 45
CN2= 169+324-234√2
CN= 12.7308 km.
The total distance of the race would be 13+18+12.7=43.7 km.
b)
I already know the angle of Eastdale which is 45° due to the turn from due east to the northwest. So therefore so I need to find the measures of angle c and angle n. I can use the Law of Sines.
sin‹C/c=sin‹E/e
sin‹C/18=(√2/2)/12.7308
sin‹C=18(√2/2)/12.708=.9998
angle c= 88.8°
So therefore m‹N= 180°-45°-88.8°=46.2°
Just mentioning the 45 degree angle should have satisfied the teacher, since its sin and cos are both √2/2
IMO
To calculate √2/2, you can use the unit circle or a calculator.
The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) on a coordinate plane. By drawing a line from the origin to a point on the circle, the x-coordinate of that point is equal to the cosine of the angle formed by the line and the positive x-axis, and the y-coordinate is equal to the sine of the angle.
Since √2/2 is a ratio of two sides in a right triangle, it represents the y-coordinate of a point on the unit circle formed by a 45-degree angle. In this case, √2/2 represents the value of sine (sin) for a 45-degree angle.
To find the value of √2/2, you can either use a calculator or use the special angles in the unit circle.
For example, at 45 degrees, the coordinates on the unit circle are (√2/2, √2/2). Therefore, the value of √2/2 is 0.7071 when rounded to four decimal places.