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Name ___________________________ 10-15 Notes IB Math SL Review day 2- identities Prove each identity. 1. ( sin θ+cos θ ) 2 =1 + sin2 θ 2. cos 4 θ−sin 4 θ=cos2 θ 3. sin 2 θ= 1 2 − 1 2 cos 2 θ

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Page 1: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

Name ___________________________ 10-15 NotesIB Math SL

Review day 2- identities

Prove each identity.

1. (sin θ+cos θ )2=1+sin 2 θ

2. cos4 θ−sin4 θ=cos2θ

3. sin2 θ=12−1

2cos2θ

Page 2: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

4. cos2θ=12+ 1

2cos2θ

5.sin2 x−sin x

cos2x−cos x+1=tan x

6. If A is an obtuse angle in a triangle and sin A = 135

, calculate the exact value of sin 2A.

(Total 4 marks)

Angles greater than 90 but less than 180

Page 3: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

7. (a) Express 2 cos2 x + sin x in terms of sin x only.

(b) Solve the equation 2 cos2 x + sin x = 2 for x in the interval 0 x , giving your answers exactly.

(Total 4 marks)

8. (a) Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c.

(b) Hence or otherwise, solve the equation

3 sin2 x + 4 cos x – 4 = 0, 0 x 90.

(Total 4 marks)

Page 4: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

9. Given that sin x = 31

, where x is an acute angle, find the exact value of

(a) cos x;

(b) cos 2x.

(Total 6 marks)

10. Consider the trigonometric equation 2 sin2 x = 1 + cos x.

(a) Write this equation in the form f (x) = 0, where f (x) = a cos2 x + b cos x + c,and a, b, c .

(a) Factorize f (x).

(d) Solve f (x) = 0 for 0° x 360°.

(Total 6 marks)

Page 5: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

11. The following diagram shows a triangle ABC, where BCA is 90, AB = 3, AC = 2 and B A C is .

(a) Show that sin = 35

.

(b) Show that sin 2 = 954

.

(c) Find the exact value of cos 2.

(Total 6 marks)

Page 6: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

Extra Problems-optional

Prove each identity.

1. cosθ

1+sinθ+ 1+sin θ

cos θ= 2

cosθ

2. (1+ 1cosθ )¿

Page 7: bergmath.files.wordpress.com · Web viewReview day 2- identities Prove each identity. sin θ + cos θ 2 =1+ sin 2θ cos 4 θ - sin 4 θ = cos 2θ sin 2 θ = 1 2 - 1 2 cos 2θ cos

3. (a) Consider the equation 4x2 + kx + 1 = 0. For what values of k does this equation have two equal roots?

(3)

4. Let f be the function f ( ) = 2 cos 2 + 4 cos + 3, for −360 360.

(b) Show that this function may be written as f ( ) = 4 cos2 + 4 cos + 1.

(1)

(c) Consider the equation f ( ) = 0, for −360 360.

(i) How many distinct values of cos satisfy this equation?

(ii) Find all values of which satisfy this equation.

(5)


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