## Challenge 344: Some Length-y Trigonometry

(i) Use a calculator to check that sin 15^{o} + √2 = √3 cos 15^{o}

(ii) Triangle PQR has a right-angle at R, and the angle at Q is 60^{o}.

Triangle PQS is a right-angled isosceles triangle, with the right-angle at S.

S is the same side of PQ as R, and QS crosses PR at the point T.

Using a diagram of these two triangles, or otherwise, prove that sin 15^{o} + √2 = √3 cos 15^{o} without using a calculator or other technology.

### Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.