Given one function find others

1 The acute angle \(x\) radians is such that \(\tan x = k\), where \(k\) is a positive constant. Express, in terms of \(k\),

1. \(\tan ( \pi - x )\),
2. \(\tan \left( \frac { 1 } { 2 } \pi - x \right)\),
3. \(\sin x\).

3 The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).

1. Find an expression, in terms of \(k\), for
   1. \(\sin \theta\),
   2. \(\tan \theta\).
   3. Explain why \(\sin 2 \theta\) is negative for \(0 < k < 1\).

2 Given that \(x = \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)\), find the exact value of

1. \(\cos ^ { 2 } x\),
2. \(\tan ^ { 2 } x\).

5. The angle \(x\) and the angle \(y\) are such that $$\tan x = m \text { and } 4 \tan y = 8 m + 5$$ where \(m\) is a constant.
Given that \(16 \sec ^ { 2 } x + 16 \sec ^ { 2 } y = 537\)

1. find the two possible values of \(m\). Given that the angle \(x\) and the angle \(y\) are acute, find the exact value of
2. \(\sin x\)
3. \(\cot y\)

3. Given \(\cos \theta ^ { \circ } = p\), where \(p\) is a constant and \(\theta ^ { \circ }\) is acute use standard trigonometric identities to find, in terms of \(p\),

1. \(\sec \theta ^ { \circ }\)
2. \(\sin ( \theta - 90 ) ^ { \circ }\)
3. \(\sin 2 \theta ^ { \circ }\) Write each answer in its simplest form.

2. Given that \(\tan 40 ^ { \circ } = p\), find in terms of \(p\)

1. \(\cot 40 ^ { \circ }\)
2. \(\sec 40 ^ { \circ }\)
3. \(\tan 85 ^ { \circ }\)

2 It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac { 12 } { 13 }\). Find the exact value of

1. \(\cot \theta\),
2. \(\cos 2 \theta\).

5

1. Express \(\tan 2 \alpha\) in terms of \(\tan \alpha\) and hence solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation $$\tan 2 \alpha \tan \alpha = 8 .$$
2. Given that \(\beta\) is the acute angle such that \(\sin \beta = \frac { 6 } { 7 }\), find the exact value of
   1. \(\operatorname { cosec } \beta\),
   2. \(\cot ^ { 2 } \beta\).

5 In Fig. 5, triangles \(\mathrm { ABC } , \mathrm { ACD }\) and ADE are all right-angled, and angles \(\mathrm { BAC } , \mathrm { CAD }\) and DAE are all \(\theta\). \(\mathrm { AB } = x\) and \(\mathrm { AE } = 2 x\). \begin{figure}[h] \end{figure}

1. Show that \(\sec ^ { 3 } \theta = 2\).
2. Hence show the ratio of lengths ED to CB is \(2 ^ { \frac { 2 } { 3 } } : 1\).

4 The acute angles \(\alpha\) and \(\beta\) are such that $$2 \cot \alpha = 1 \text { and } 24 + \sec ^ { 2 } \beta = 10 \tan \beta \text {. }$$

1. State the value of \(\tan \alpha\) and determine the value of \(\tan \beta\).
2. Hence find the exact value of \(\tan ( \alpha + \beta )\).

2 Using an appropriate identity in each case, find the possible values of

1. \(\sin \alpha\) given that \(4 \cos 2 \alpha = \sin ^ { 2 } \alpha\),
2. \(\sec \beta\) given that \(2 \tan ^ { 2 } \beta = 3 + 9 \sec \beta\).

2 It is given that \(\theta\) is the acute angle such that \(\cot \theta = 4\). Without using a calculator, find the exact value of

1. \(\tan \left( \theta + 45 ^ { \circ } \right)\),
2. \(\operatorname { cosec } \theta\).

4 It is given that \(A\) and \(B\) are angles such that $$\sec ^ { 2 } A - \tan A = 13 \quad \text { and } \quad \sin B \sec ^ { 2 } B = 27 \cos B \operatorname { cosec } ^ { 2 } B$$ Find the possible exact values of \(\tan ( A - B )\).

1 In this question you must show detailed reasoning.
The obtuse angle \(\theta\) is such that \(\sin \theta = \frac { 2 } { \sqrt { 13 } }\).
Find the exact value of \(\cos \theta\).

3 In this question you must show detailed reasoning.

1. Given that \(\sin \alpha = \frac { 2 } { 3 }\), find the exact values of \(\cos \alpha\).
2. Given that \(2 \tan ^ { 2 } \beta - 7 \sec \beta + 5 = 0\), find the exact value of \(\sec \beta\).

It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac{12}{13}\). Find the exact value of

1. \(\cot \theta\), [2]
2. \(\cos 2\theta\). [3]