CAIE P1 2022 November — Question 6 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2022
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTrig Graphs & Exact Values
TypeFind exact trig values from given ratio
DifficultyModerate -0.3 This is a straightforward application of Pythagorean identity and trigonometric definitions. Given cos α = 8/17, students find sin α = 15/17 using sin²α + cos²α = 1, then tan α = 15/8, and substitute into the expression. It requires basic manipulation but no problem-solving insight, making it slightly easier than average.
Spec1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1
6 It is given that \(\alpha = \cos ^ { - 1 } \left( \frac { 8 } { 17 } \right)\).
Find, without using the trigonometric functions on your calculator, the exact value of \(\frac { 1 } { \sin \alpha } + \frac { 1 } { \tan \alpha }\).
Question 6:
AnswerMarks Guidance
Use of \(\sin^2\alpha + \cos^2\alpha = 1\), e.g. \(\sin\alpha = [\pm]\sqrt{1 - \left(\frac{8}{17}\right)^2}\)\*M1 Or Pythagoras seen (may quote 8, 15, 17 triple).
\(\sin\alpha = \frac{15}{17}\)A1
\(\tan\alpha = \frac{15}{8}\)A1
\(\frac{1}{\sin\alpha} + \frac{1}{\tan\alpha} = \frac{17}{15} + \frac{8}{15}\)DM1 Dealing with reciprocals and addition of fractions correctly.
\(= \frac{5}{3}\) oeA1 Correct answer with no working shown scores 0. Extra answers from \(\sin\alpha = -\frac{15}{17}\) are allowed. 
## Question 6:

Use of $\sin^2\alpha + \cos^2\alpha = 1$, e.g. $\sin\alpha = [\pm]\sqrt{1 - \left(\frac{8}{17}\right)^2}$ | **\*M1** | Or Pythagoras seen (may quote 8, 15, 17 triple).

$\sin\alpha = \frac{15}{17}$ | **A1** |

$\tan\alpha = \frac{15}{8}$ | **A1** |

$\frac{1}{\sin\alpha} + \frac{1}{\tan\alpha} = \frac{17}{15} + \frac{8}{15}$ | **DM1** | Dealing with reciprocals and addition of fractions correctly.

$= \frac{5}{3}$ oe | **A1** | Correct answer with no working shown scores 0. Extra answers from $\sin\alpha = -\frac{15}{17}$ are allowed.

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6 It is given that $\alpha = \cos ^ { - 1 } \left( \frac { 8 } { 17 } \right)$.\\
Find, without using the trigonometric functions on your calculator, the exact value of $\frac { 1 } { \sin \alpha } + \frac { 1 } { \tan \alpha }$.\\

\hfill \mbox{\textit{CAIE P1 2022 Q6 [5]}}
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