Given sin/cos/tan, find other expressions

6 It is given that \(3 \sin 2 \theta = \cos \theta\) where \(\theta\) is an angle such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

1. Find the exact value of \(\sin \theta\).
2. Find the exact value of \(\sec \theta\).
3. Find the exact value of \(\cos 2 \theta\).

1 It is given that \(\theta\) is an acute angle in degrees such that \(\sin \theta = \frac { 2 } { 3 }\).
Find the exact value of \(\sin \left( \theta + 60 ^ { \circ } \right)\).

3 It is given that \(\cos a = \frac { 3 } { 5 }\), where \(0 ^ { \circ } < a < 90 ^ { \circ }\). Showing your working and without using a calculator to evaluate \(a\),

1. find the exact value of \(\sin \left( a - 30 ^ { \circ } \right)\),
2. find the exact value of \(\tan 2 a\), and hence find the exact value of \(\tan 3 a\).

1

1. Simplify \(\sin 2 \alpha \sec \alpha\).
2. Given that \(3 \cos 2 \beta + 7 \cos \beta = 0\), find the exact value of \(\cos \beta\).

1. (a) By writing \(\sin 3 \theta\) as \(\sin ( 2 \theta + \theta )\), show that

$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (b) Given that \(\sin \theta = \frac { \sqrt { } 3 } { 4 }\), find the exact value of \(\sin 3 \theta\).

1. Given that

$$\tan \theta ^ { \circ } = p , \text { where } p \text { is a constant, } p \neq \pm 1$$ use standard trigonometric identities, to find in terms of \(p\),

1. \(\tan 2 \theta ^ { \circ }\)
2. \(\cos \theta ^ { \circ }\)
3. \(\cot ( \theta - 45 ) ^ { \circ }\) Write each answer in its simplest form.

4. (i) Given that \(\cos x = \sqrt { 3 } - 1\), find the value of \(\cos 2 x\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
(ii) Given that $$2 \cos ( y + 30 ) ^ { \circ } = \sqrt { 3 } \sin ( y - 30 ) ^ { \circ }$$ find the value of \(\tan y\) in the form \(k \sqrt { 3 }\) where \(k\) is a rational constant.

5

1. Write down the identity expressing \(\sin 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
2. Given that \(\sin \alpha = \frac { 1 } { 4 }\) and \(\alpha\) is acute, show that \(\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }\).
3. Solve, for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\), the equation \(5 \sin 2 \beta \sec \beta = 3\).

2 The angle \(\theta\) is such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

1. Given that \(\theta\) satisfies the equation \(6 \sin 2 \theta = 5 \cos \theta\), find the exact value of \(\sin \theta\).
2. Given instead that \(\theta\) satisfies the equation \(8 \cos \theta \operatorname { cosec } ^ { 2 } \theta = 3\), find the exact value of \(\cos \theta\).

9 The value of \(\tan 10 ^ { \circ }\) is denoted by \(p\). Find, in terms of \(p\), the value of

1. \(\tan 55 ^ { \circ }\),
2. \(\tan 5 ^ { \circ }\),
3. \(\tan \theta\), where \(\theta\) satisfies the equation \(3 \sin \left( \theta + 10 ^ { \circ } \right) = 7 \cos \left( \theta - 10 ^ { \circ } \right)\).

3

1. Given that \(7 \sin 2 \alpha = 3 \sin \alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), find the exact value of \(\cos \alpha\).
2. Given that \(3 \cos 2 \beta + 19 \cos \beta + 13 = 0\), where \(90 ^ { \circ } < \beta < 180 ^ { \circ }\), find the exact value of \(\sec \beta\).

3 It is given that \(\theta\) is the acute angle such that \(\sec \theta \sin \theta = 36 \cot \theta\).

1. Show that \(\tan \theta = 6\).
2. Hence, using an appropriate formula in each case, find the exact value of
   (a) \(\tan \left( \theta - 45 ^ { \circ } \right)\),
   (b) \(\quad \tan 2 \theta\).

8. (a) Given that \(\sin \theta = 2 - \sqrt { 2 }\), find the value of \(\cos ^ { 2 } \theta\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
(b) Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos \left( 2 x - \frac { \pi } { 6 } \right) = \frac { 1 } { 2 } .$$

1. (a) Given that \(\cos x = \sqrt { 3 } - 1\), find the value of \(\cos 2 x\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
   (b) Given that

$$2 \cos ( y + 30 ) ^ { \circ } = \sqrt { 3 } \sin ( y - 30 ) ^ { \circ }$$ find the value of \(\tan y\) in the form \(k \sqrt { 3 }\) where \(k\) is a rational constant.

2 Angle \(\alpha\) is acute and \(\cos \alpha = \frac { 3 } { 5 }\). Angle \(\beta\) is obtuse and \(\sin \beta = \frac { 1 } { 2 }\).

1. 1. Find the value of \(\tan \alpha\) as a fraction.
      (1 mark)
   2. Find the value of \(\tan \beta\) in surd form.
2. Hence show that \(\tan ( \alpha + \beta ) = \frac { m \sqrt { 3 } - n } { n \sqrt { 3 } + m }\), where \(m\) and \(n\) are integers.
   (3 marks)

2 The acute angles \(\alpha\) and \(\beta\) are given by \(\tan \alpha = \frac { 2 } { \sqrt { 5 } }\) and \(\tan \beta = \frac { 1 } { 2 }\).

1. 1. Show that \(\sin \alpha = \frac { 2 } { 3 }\), and find the exact value of \(\cos \alpha\).
   2. Hence find the exact value of \(\sin 2 \alpha\).
2. Show that the exact value of \(\cos ( \alpha - \beta )\) can be expressed as \(\frac { 2 } { 15 } ( k + \sqrt { 5 } )\), where \(k\) is an integer.

5 It is given that \(\sin A = \frac { \sqrt { 5 } } { 3 }\) and \(\sin B = \frac { 1 } { \sqrt { 5 } }\), where the angles \(A\) and \(B\) are both acute.

1. 1. Show that the exact value of \(\cos B = \frac { 2 } { \sqrt { 5 } }\).
   2. Hence show that the exact value of \(\sin 2 B\) is \(\frac { 4 } { 5 }\).
2. 1. Show that the exact value of \(\sin ( A - B )\) can be written as \(p ( 5 - \sqrt { 5 } )\), where \(p\) is a rational number.
   2. Find the exact value of \(\cos ( A - B )\) in the form \(r + s \sqrt { 5 }\), where \(r\) and \(s\) are rational numbers.

5

1. The angle \(\alpha\) is acute and \(\sin \alpha = \frac { 4 } { 5 }\).
   1. Find the value of \(\cos \alpha\).
   2. Express \(\cos ( \alpha - \beta )\) in terms of \(\sin \beta\) and \(\cos \beta\).
   3. Given also that the angle \(\beta\) is acute and \(\cos \beta = \frac { 5 } { 13 }\), find the exact value of \(\cos ( \alpha - \beta )\).
2. 1. Given that \(\tan 2 x = 1\), show that \(\tan ^ { 2 } x + 2 \tan x - 1 = 0\).
   2. Hence, given that \(\tan 45 ^ { \circ } = 1\), show that \(\tan 22 \frac { 1 } { 2 } ^ { \circ } = \sqrt { 2 } - 1\).