Moderate -0.8 This is a straightforward application of the Pythagorean identity using a right-angled triangle or the identity 1 + tan²θ = sec²θ. Given tan θ = 1/2, students construct a triangle with opposite = 1, adjacent = 2, hypotenuse = √5, then cos θ = 2/√5, so cos²θ = 4/5. It's simpler than average A-level questions as it requires only one standard technique with no problem-solving or multi-step reasoning.
7 You are given that \(\tan \theta = \frac { 1 } { 2 }\) and the angle \(\theta\) is acute. Show, without using a calculator, that \(\cos ^ { 2 } \theta = \frac { 4 } { 5 }\).
Right angled triangle with 1 and 2 on correct sides
M1
Pythagoras used to obtain hyp \(= \sqrt{5}\)
M1
\(\cos \theta = \frac{2}{\sqrt{5}}\)
A1
*or* M1 for \(\sin\theta = \frac{1}{2}\cos\theta\) and M1 for substituting in \(\sin^2 \theta + \cos^2\theta = 1\)
Answer
Marks
E1 for sufficient working
3
**Question 7:**
Right angled triangle with 1 and 2 on correct sides | M1
Pythagoras used to obtain hyp $= \sqrt{5}$ | M1
$\cos \theta = \frac{2}{\sqrt{5}}$ | A1
*or* M1 for $\sin\theta = \frac{1}{2}\cos\theta$ and M1 for substituting in $\sin^2 \theta + \cos^2\theta = 1$
E1 for sufficient working | 3
7 You are given that $\tan \theta = \frac { 1 } { 2 }$ and the angle $\theta$ is acute. Show, without using a calculator, that $\cos ^ { 2 } \theta = \frac { 4 } { 5 }$.
\hfill \mbox{\textit{OCR MEI C2 Q7 [3]}}