Grade 10 Trigonometry
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1 ID : ww-10-trigonometry [1] Grade 10 Trigonometry For more such worksheets visit Answer t he quest ions (1) If - 0, f ind value of sin 4 θ - cos 4 θ. (2) Simplif y 3(sin 4 θ cos 4 θ) - 2(sin 6 θ cos 6 θ) (3) Simplif y (sin 4 α - cos 4 α 1) cosec 2 α (4) Simplif y Choose correct answer(s) f rom given choice (5) Simplif y. a. 1 tanθ cotθ b. cos 2 θ - sin 2 θ c. - d. sin 2 θ - cos 2 θ (6)? a. cosec θ tan θ b. sec θ cot θ c. sec θ - tan θ d. cosec θ cot θ (7) If α β 90, simplif y. a. ± cos α b. 0 c. 1 d. ± sin α (8)? a. b. - c. 0 d. 2 (9)? a. 1 tanθ b. tanθ - cot θ c. tanθ cot θ d. 1 cot θ
2 (10) If AB and BC, f ind value of (cos A / tan C). ID : ww-10-trigonometry [2] a. b. c. d. (11) Simplif y sin 6 β cos 6 β 3sin 2 β cos 2 β a. 1 b. 3 c. 0 d. 2 (12) If sin(ab) 1, cos(a-b)? a. sin B b. cos 2A c. sin 2B d. cos A Fill in the blanks (13) sin 2 71 sin 2 19 cos 2 71 cos 2 19 sin 2 19 sin 71 cos 19. Check True/False (14) If the length of the shadow of a building is increasing, then the angle of elevation of the sun is decreasing. True False (15) sin 43 - cos 43 > 0 True False 2016 Edugain ( All Rights Reserved Many more such worksheets can be generated at
3 Answers ID : ww-10-trigonometry [3] (1) 0 It is given that - 0, which means that We also know that, cos 2 θ cos 2 θ 1/(11) cos 2 θ 1/2 We also know that, sin 2 θ 1/(11) sin 2 θ 1/2 Step 4 Theref ore, sin 4 θ - cos 4 θ sin 4 θ - cos 4 θ 0 (2) 1 3 (sin 4 θ cos 4 θ) - 2 [ (sin 2 θ) 3 (cos 2 θ) 3 } ] 3 (sin 4 θ cos 4 θ) - 2 [ (sin 2 θ cos 2 θ) {(sin 2 θ) 2 (cos 2 θ) 2 - sin 2 θ cos 2 θ } ] Since sin 2 θ cos 2 θ 1 3 (sin 4 θ cos 4 θ) - 2 {sin 4 θ cos 4 θ - sin 2 θ cos 2 θ } Step 4 sin 4 θ cos 4 θ 2 sin 2 θ cos 2 θ Step 5 (sin 2 θ cos 2 θ)
4 (3) 2 ID : ww-10-trigonometry [4] Lets f irst simplif y sin 4 α - cos 4 α 1 S sin 4 α - cos 4 α 1 S sin 4 α - (cos 2 α) 2 1 S sin 4 α - (1 - sin 2 α) [Using identity cos 2 θ 1 sin 2 θ] S sin 4 α - (1 sin 4 α - 2 sin 2 α) 1... [Using identity (a - b) 2 a 2 b 2-2ab] S sin 4 α sin 4 α 2 sin 2 α 1 S sin 4 α sin 4 α 2 sin 2 α) 1 S 2 sin 2 α Theref ore (sin 4 α - cos 4 α 1) cosec 2 α 2 sin 2 α cosec 2 α 2 (4) 2 cosecθ
5 (5) a. 1 tanθ cotθ ID : ww-10-trigonometry [5] We know that, tanθ, cotθ Now, cotθ 1 - tanθ tanθ 1 - cotθ can be simplif ied as: cotθ 1 - tanθ tanθ 1 - cotθ cos 2 θ ( - ) sin 2 θ ( - ) cos 3 θ - sin 3 θ.( - ) ( - )(cos2 θ. sin 2 θ).( - ) cos2 θ. sin 2 θ. cos 2 θ... sin 2 θ. 1 cotθ 1 tanθ 1 tanθ cotθ
6 (6) d. cosec θ cot θ ID : ww-10-trigonometry [6] We know that 1 - cos 2 θ sin 2 θ. Using this identity we can simplif y denominator, if we multiply it by 1 Now on multiplying numerator and denominator by 1, ( ) ( ) ( ( (1 )2 1 - cos 2 θ (1 )2 sin 2 θ ) ) 1 1 cosecθ cotθ (7) d. ± sin α It is given that α β 90, theref ore β 90 - α Replace β with 90 - α in given expression S S Using trigonometrical identities S S S ± sin α
7 ID : ww-10-trigonometry [7] (8) d. 2 Since sin(90 -θ) cos θ and cos(90 -θ) sin θ, expression can be rewritten as f ollowing cos2 20 sin 2 20 sin 2 20 cos 2 20 tanθ cotθ 1 cos 2 θ sin 2 θ (9) c. tanθ cot θ Using identities sec 2 θ 1 tan 2 θ and cosec 2 θ 1 cot 2 θ S S Using the relation tanθcotθ 1, we can re-write above expression as f ollowing S Using (a b) 2 a 2 b 2 2ab S S tanθ cotθ (10) d. AC cos A / tan C (AB/AC) / (AB/BC)
8 (11) a. 1 ID : ww-10-trigonometry [8] S sin 6 β cos 6 β 3sin 2 β cos 2 β S (sin 2 β) 3 (cos 2 β) 3 3sin 2 β 3sin 2 β cos 2 β S (sin 2 β cos 2 β) [(sin 2 β) 2 - sin 2 β cos 2 β (cos 2 β) 2 ] 3sin 2 β cos 2 β... Using a 3 b 3 (ab)(a 2 - ab b 2 ) Step 4 S 1[(sin 2 β) 2 - sin 2 β cos 2 β (cos 2 β) 2 ] 3sin 2 β cos 2 β... Using sin 2 θ cos 2 θ 1 Step 5 S (sin 2 β) 2 2sin 2 β cos 2 β (cos 2 β) 2 Step 6 S (sin 2 β cos 2 β) 2 ) 2... Using a 2 b 2 2ab (a b) 2 Step 7 S Using sin 2 θ cos 2 θ 1 S 1 (12) c. sin 2B We are given that sin(a B) 1, but we already know that sin 90 1 This means, A B 90 > A 90 - B...(1) Now replace value of A f rom eq. (1) in cos(a-b) : cos(a-b) cos(90 - B - B) cos(90-2b) sin(2b)... [ Here we have applied the identity cos(90 - θ) sin(θ) ]
9 (13) 2 ID : ww-10-trigonometry [9] Lets, S sin 2 71 sin 2 19 cos 2 71 cos 2 19 sin 2 19 sin 71 cos 19 Using identities sin(θ) cos(90 - θ) and cos(θ) sin(90 - θ), we can re-write expression as f ollowing S sin 2 71 cos 2 (90-19 ) cos 2 71 cos 2 (90-19 ) sin 2 19 cos (90-71 ) cos 19 S sin 2 71 cos 2 71 cos 2 71 cos 2 71 sin 2 19 cos 19 cos 19 Using identity sin 2 θ cos 2 θ 1 S 1/1 sin 2 19 cos 2 19 S 1 1 S 2
10 (14) True ID : ww-10-trigonometry [10] Let's assume, BD is the lengths of the shadow of the building AB and the length of the shadow of the building is increasing f rom BD to BC, as shown in the f ollowing f igure, It can be seen that angle x > y, theref ore as the shadow of the building is increasing, the angle of elevation of the sun is decreasing Theref ore, given statement is True. (15) False We know that is greater than, if θ < 45. Theref ore sin 43 - cos 43 will be negative, and statement "sin 43 - cos 43 > 0" is False.
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