Trigonometry ext 1 maths questions (1 Viewer)

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Hey guys! I have a test coming up and I would love it if someone could help me with these questions that I don't understand how to do, thank you!
1. One vertex angle of a rhombus is 25degrees. Find the ratio of the diagonals. Answer to 4 significant figures.
2. From a ship sailing due north, a lighthouse is observed to be on a bearing of 42degrees. Later, when the ship is 2 nautical miles from the lighthouse, the bearing of the lighthouse from the ship is 148degrees. Find, correct to three sig figs, the distance of the lighthouse from the initial point of observation.
3. PQR is an equilateral triangle of side length x, and PS is the perpendicular from P to QR. PS is produced to T so that PT = x
a) Show that angle PQT = 75degrees and hence that angle SQT = 15degrees
b) Show that QS = 1/2x and that PS = 1/2x(root3)
c) Show that ST = 1/2x(2-root3)
d) Hence show that tan15degrees = 2-root3
4. Prove (sin120/tan300) - (cos240/cot315) = tansquared240 - cosecsquared330
5. If x=acostheta - bsintheta and y=asintheta + bcostheta, show that xsquared + ysquared = asquared + bsquared
6. Eliminate theta from each pair of equations: x = sintheta + costheta and y = sintheta - costheta
If you help me, you're a lifesaver!! thanks guys

 

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please somebody helpppp

 

[Drongoski]

Q1

I think, bearing in mind the 2 diagonals bisect each other at right angles, and partition the rhombus into 4 congruent right angled triangles:

ratio of diagonals = ratio of semi-diagonals = tan (180 - 25)/2 = tan 77.5 deg = 4.510708... = 4.511 : 1