sin x = squareroot (1- (-4/squareroot 17)^2)

Question

the expamle shows it going right to squareroot (1/17) then 1/square root 17 from there.

 

I am not understandiung how they when from the original equation down to the squareroot (1/17)

Answer 1

(-4/√17)²=16/17.

sin(x)=√(1-16/17)=√(1/17)=1/√17, which can be rationalised to √17/17 by multiplying top and bottom by √17.

sin²(x)+cos²(x)=1, so sin(x)=√(1-cos²(x)).

Comparing this with the given equation we can see that cos(x)=-4/√17.

To find x it would appear that we can use either the cosine or the sine.

If we use the sine, then sin(x)=√17/17, x=14.04° approx.

But if we use the cosine, then cos(x)=-4/√17=-4√17/17, x=165.96° approx.

165.96°=180°-14.04°, so it would be reasonable to assume that x=165.96°, because its sine is the same as sin(14.04°). The cosine of 14.04° is 4/√17, not -4/√17. (-4/√17)² and (4/√17)² are each equal to 16/17. The reason for choosing 165.96° instead of 14.04° follows from the trig ratios of angles in different quadrants. In the first quadrant (0-90°) all the trig ratios are positive. In the second quadrant (90°-180°), only sine is positive; in the third (180°-270°) only tangent is positive; in the fourth (270°-360°) only cosine is positive. A negative cosine means the angle must be in either the 2nd or 3rd quadrant; the positive sine means the angle must be in the 1st or 2nd quadrant. Since we have a negative cosine and a positive sine the angle must be in the 2nd quadrant.