| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Standard +0.3 This is a straightforward application of standard trigonometric identities. Part (i) requires the compound angle formula for sin(a-30°) after finding sin(a) from the Pythagorean identity. Part (ii) uses the double angle formula for tan(2a), then applies the addition formula tan(3a)=tan(2a+a). All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(\sin a = 4/5\) | B1 | |
| Use \(\sin(A - B)\) formula and substitute for \(\cos a\) and \(\sin a\) | M1 | |
| Obtain answer \(\frac{1}{10}(4\sqrt{3} - 3)\), or exact equivalent | A1 | [3] |
| (ii) Use \(\tan 2A\) formula and substitute for \(\tan a\), or use \(\sin 2A\) and \(\cos 2A\) formulae, substitute \(\sin a\) and \(\cos a\), and divide | M1 | |
| Obtain \(\tan 2a = -\frac{24}{7}\), or equivalent | A1 | |
| Use \(\tan(A + B)\) formula with \(A = 2a\), \(B = a\) and substitute for \(\tan 2a\) and \(\tan a\) | M1 | |
| Obtain \(\tan 3a = -\frac{44}{117}\) | A1 | [4] |
**(i)** State or imply $\sin a = 4/5$ | B1 |
Use $\sin(A - B)$ formula and substitute for $\cos a$ and $\sin a$ | M1 |
Obtain answer $\frac{1}{10}(4\sqrt{3} - 3)$, or exact equivalent | A1 | [3]
**(ii)** Use $\tan 2A$ formula and substitute for $\tan a$, or use $\sin 2A$ and $\cos 2A$ formulae, substitute $\sin a$ and $\cos a$, and divide | M1 |
Obtain $\tan 2a = -\frac{24}{7}$, or equivalent | A1 |
Use $\tan(A + B)$ formula with $A = 2a$, $B = a$ and substitute for $\tan 2a$ and $\tan a$ | M1 |
Obtain $\tan 3a = -\frac{44}{117}$ | A1 | [4]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 $a$,\\
(i) find the exact value of $\sin \left( a - 30 ^ { \circ } \right)$,\\
(ii) find the exact value of $\tan 2 a$, and hence find the exact value of $\tan 3 a$.
\hfill \mbox{\textit{CAIE P3 2010 Q3 [7]}}