Woo | 2010 . . Trigonometry | Review Worksheet 1. Solving trigonometric equations Solve each of the equations below for the domain 0° ݔ 360° : ሺaሻ 2 sin ଶ ݔെ1ൌ0 ሺbሻ 2 tan ଶ ݔ 3 tan ݔെ2ൌ0 ሺcሻ sin 2 ݔൌ1 ሺdሻ ሺ2 sin ݔ 1ሻሺsin ݔെ 2ሻ ൌ 0 ሺeሻ 3 sin ݔെ 5 cos ݔൌ0 2. Trigonometric identities ሺaሻ Simplify: ሺ1 െ cos ଶ ܣሻሺ1 cot ଶ ܣሻ Prove the following: ሺbሻ ୱ୧୬ ଵାୡ୭ୱ ଵାୡ୭ୱ ୱ୧୬ ൌ 2 cosec ܣሺcሻ ሺcot ܣ tan ܣሻ ଶ ሺsin ܣ cosec ܣሻ ଶ ሺcos ܣ sec ܣሻ ଶ ൌ 5 2 sec ଶ ܣcosec ଶ ܣ3. If sin ܣൌ ଶ ଷ and cos ܣ 0, find the exact value of cos ܣand cot .ܣ4. A parallelogram has adjacent sides of lengths 3cm and 4cm that enclose an angle of 60 degrees. Find, in irrational form, the: ሺaሻ Area of the parallelogram; ሺbሻ Length of the shorter diagonal. 5. The lengths of the sides of a triangle are in the ratio 2 to 3 to 4. Find the size of the smallest angle. 6. The diagram below was sketched by a surveyor, who measured the angle of elevation of a tree top on the other side of a river to be 7°12Ԣ at the point .ܣAt the point ,ܤ100 metres directly towards the tree from ,ܣthe angle of elevation was 9°42Ԣ. ሺaሻ Derive an expression for the height of the tree. ሺbሻ Calculate the height of the tree, correct to three significant figures. * ܤ ܣ7°12Ԣ 9°42Ԣ NOT TO SCALE 100m
Solve each of the equations below for the domain 0° 360° : a 2 sin 1 0 b 2 tan 3 tan 2 0 c sin 2 1 d 2 sin 1 sin 2 0 e 3 sin 5 cos 0
2. Trigonometric identities a Simplify: 1 cos 1 cot Prove the following:
b 2 cosec
c cot tan sin cosec cos sec 5 2 sec cosec
3. If sin and cos 0, find the exact value of cos and cot .
4. A parallelogram has adjacent sides of lengths 3cm and 4cm that enclose an angle of 60 degrees. Find, in
irrational form, the: a Area of the parallelogram; b Length of the shorter diagonal.
5. The lengths of the sides of a triangle are in the ratio 2 to 3 to 4. Find the size of the smallest angle.
6. The diagram below was sketched by a surveyor, who measured the angle of elevation of a tree top on the other side of a river to be 7°12 at the point . At the point , 100 metres directly towards the tree from , the angle of elevation was 9°42 . a Derive an expression for the height of the tree. b Calculate the height of the tree, correct to three significant figures.
*
7°12 9°42
NOT TO SCALE
100m
Woo | 2010
7. Two artillery guns are situated 3 kilometres apart in a straight line at positions and , where is due east of . They are both aiming at a target, . The bearing of from is 018°T and from is 282°T. Find the distance between the target and the gun nearer to it.
8. A ship steaming due east observes a lighthouse in a direction of N50°E and after the ship has steamed 3 nautical miles the lighthouse is N30°E. How much farther must the ship proceed until the lighthouse bears due north and what will then be the distance between them?
9. A man observes at the top of a distant peak has an angle of elevation of 24°. After advancing a distance of 2
kilometres up a path, inclined at 8° to the horizontal, directly towards the peak, he finds the angle of elevation then to the peak to be 28°. Find the height of the peak above his first point of elevation, and also the horizontal distance of the peak from this point.
10. At two points and , in the same horizontal plane, a balloon has angles of elevation of 37° and 25°
respectively. is due south of the balloon and is east of . Find the height of the balloon, to the nearest metre, if the distance between and is 800 metres.
11. is a rectangle in the horizontal plane. At a point vertically below , the angles of elevation of , and
are , and respectively. Prove that cot cot cot .
12. If , show that sin cos .
13. In ∆ , is the point on such that .
a Using the sine rule, or otherwise, show that : : . b Express the area of ∆ in terms of .
c By considering areas of triangles, or otherwise show that if 60°, then .
