Year 10 Term 1 Homework

Transcription

1 Yimin Math Centre Year 10 Term 1 Homework Student Name: Grade: Date: Score: Table of contents 6 Year 10 Term 1 Week 6 Homework Triangle trigonometry The tangent ratio The complementary results Trigonometric rations of obtuse angles The exact values The Sine Rule The Cosine Rule Miscellaneous exercises This edition was printed on October 24, 2016 with worked solutions. Camera ready copy was prepared with the L A TEX2e typesetting system. Year 10 Term 1 Homework

2 Year 10 Term 1 Week 6 Homework Page 1 of 10 6 Year 10 Term 1 Week 6 Homework 6.1 Triangle trigonometry The tangent ratio The tangent ratio can be expressed as the quotient of sine and cosine ratios. tanθ = sinθ cosθ Exercise Express each equation o terms of tanθ, then solve for θ, correct to the nearest minute cos θ = 5 sin θ 2. 4 sin θ = 1 cos θ 3. 3 sin θ = 3 cos θ Exercise Prove each of the following identities: 1. sin θ cos θ tan θ = cos 2 θ 2. cos θ tan θ sin θ = 1 3. sin 2 θ+sin θ cos θ cos 2 θ+sin θ cos θ = tan θ

3 Year 10 Term 1 Week 6 Homework Page 2 of The complementary results In any right-angled triangle, the sine of an acute angle is equal to the cosine of its complement, and the cosine of an acute angle is equal to the sine of its complement. sin θ = cos(90 θ) and cos θ = sin(90 θ) Example Solve each of these equations: 1. cos(2x + 58) = sin 12 Solution: cos(2x + 58 ) = cos(90 12 ) = cos 78, 2x + 58 = 78 x = sin(x + 18) = cos(x 18) Solution: sin(x + 18 ) = sin[90 (x 18 )], x + 18 = 90 x + 18, 2x = 90 x = 45. Exercise Simplify the following expressions: 1. cos(90 θ) tan θ 2. sin(90 θ) cos(90 θ) tan(90 θ)

4 Year 10 Term 1 Week 6 Homework Page 3 of Trigonometric rations of obtuse angles Definition: If θ is an acute angle, then: sin(180 θ) = sin θ cos(180 θ) = cos θ tan(180 θ) = tan θ Exercise State whether the angle θ is acute or obtuse, where 0 < θ < 180, if: 1. sin θ > 0 and tan θ > 0 2. cos θ > 0 and tan θ > 0 3. tan θ < 0 and cos θ < 0 Exercise Express each of the following trigonometric ratios on terms of an acute angle, then evaluate, correct to 4 decimal places. 1. tan cos sin 135 Exercise For each of the following, find θ, where 0 < θ < 180. Answer correct to the nearest degree. 1. cos θ = tan θ = sin θ = 0.848

5 Year 10 Term 1 Week 6 Homework Page 4 of 10 Exercise Prove each of the following identities: 1. sin θ sin(90 θ) cos θ cos(90 θ) = 1 2. cos(180 θ) sin θ sin(180 θ) = cos θ 3. tan(180 θ) tan(90 θ) = 1 4. cos(90 θ) cos(180 θ) = tan θ 5. sin θ cos(90 θ) tan 2 θ = cos 2 θ 6. tan(180 θ) sin(90 θ) = sin θ

6 Year 10 Term 1 Week 6 Homework Page 5 of The exact values Exercise Find the exact value of each expression: 1. tan cos (sin cos 2 45 ) 3. sin 2 60 cos 2 60 Exercise Show that: 1. sin 60 cos 30 + cos 60 sin 30 = 1 2. sin 45 cos 60 + cos 45 sin 60 =

7 Year 10 Term 1 Week 6 Homework Page 6 of The Sine Rule Definition: The Sine rule states that in any triangle ABC: a sin A = b sin B = c sin C or sin A a = sin B b = sin C c Example I ABC, A = 30, BC = 18 cm and AB = 25 cm. Find the two possible values for C, correct to nearest minutes. Hence show that there are two possible triangles. Solution: sin C sin 30 sin 30 =, sin C = 25 C = sin 1 = As sin(180 θ) = sin θ. C = = 136. Therefore there are two possible triangles. Exercise Find the size of the acute angle θ, correct to the nearest minute. 2. Find the value of x correct to 1 decimal place.

8 Year 10 Term 1 Week 6 Homework Page 7 of The Cosine Rule Definition: The Cosine Rule states that in any triangle ABC: a 2 = b 2 + c 2 2bc cos A, and cos A = b2 + c 2 a 2 2bc Example Find the value of the pronumeral, correct to 1 decimal place. Solution: From the formula: q 2 = (11)(8) cos 115, q 2 = , q = = 16.1 cm. 2. Find the angle θ in the triangle given below, correct to the nearest minute. Solution: From the formula: cos θ = = (10)(7) θ = Exercise The sides of a certain triangle are in the ratio 5 : 6 : 8. Find the size of angles, correct to the nearest degree.

9 Year 10 Term 1 Week 6 Homework Page 8 of 10 Exercise Find the exact value of x in the triangle shown below: 2. In ABC, AB = BC = 6 cm. AC is produce to D so that CD = 5 cm and BD = 9 cm. Find the exact length of AC.

10 Year 10 Term 1 Week 6 Homework Page 9 of Miscellaneous exercises Example Find the exact value of CE given that AE = 16 cm. Solution: sin 30 = 1 2 = ED 16 ED = 8 = BD, cos 30 = AD, AD = AB = cos 30 = EC = = 8 3, EC = cm. Exercise In ABC, AB = 24 cm. CAB = 60 and CBA = 75. Find as exact values of AC and BC. Hence, find the area of ABC.

11 Year 10 Term 1 Week 6 Homework Page 10 of 10 Exercise Find the distance AB shown in the figure (in exact value), given that θ = Simplify 1 x x 3. Simplify 5 x 3 x x Evaluate ( 8) 2 3.