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26 \(ABC\) is a right-angled triangle.
Calculate the value of \(x\).
Give your answer correct to 1 decimal place. (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| 60.3 | M1 | for a correct trig statement, eg \(\tan x = \dfrac{7}{4}\) |
| A1 | 60.2 to 60.42 |
Other methods are possible but only award this mark at the point of an equation with \(x\) as the only unknown
7 \(ABC\) is a right-angled triangle.
Calculate the value of \(x\).
Give your answer correct to 1 decimal place. (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| 60.3 | M1 | for a correct trig statement, eg \(\tan x = \frac{7}{4}\) |
| A1 | 60.2 to 60.42 |
Other methods are possible but only award this mark at the point of an equation with \(x\) as the only unknown
24 \(ABC\) is a right-angled triangle.
Calculate the length of \(AB\).
Give your answer correct to 1 decimal place. (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| 14.3 | M1 | for a correct statement for \(AB\) using trigonometry, eg \(\tan 62 = \dfrac{AB}{7.6}\) or \((AB =)\ 7.6 \times \tan 62\) |
| A1 | answer in the range 14.28 to 14.3 |
If an answer is given in the range in working and then rounded incorrectly award full marks
8 \(ABC\) is a right-angled triangle.
Calculate the length of \(AB\).
Give your answer correct to 1 decimal place. (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| 14.3 | M1 | for a correct statement for \(AB\) using trigonometry, eg \(\tan 62 = \dfrac{AB}{7.6}\) or \((AB =)\ 7.6 \times \tan 62\) |
| A1 | answer in the range 14.28 to 14.3 |
If an answer is given in the range in working and then rounded incorrectly award full marks
26
Not drawn accurately
Use trigonometry to work out the value of \(x\).
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: uses or states tan | ||
| tan stated or used | M1 | implied by [0.726, 0.73] or [1.376, 1.4] |
| \(\tan 36 = \dfrac{x}{4}\) or \(4 \times \tan 36\) or \(\tan 54 = \dfrac{4}{x}\) or \(\dfrac{4}{\tan 54}\) | M1dep | oe accept [0.726, 0.73] for tan 36 accept [1.376, 1.4] for tan 54 |
| [2.9, 2.91] with M2 awarded | A1 | accept 3 with M2 awarded |
| Alternative method 2: works out hypotenuse first | ||
| (hypotenuse \(=\)) \(\dfrac{4}{\sin 54}\) or \(\dfrac{4}{\cos 36}\) or 4.9(…) | M1 | oe accept [0.8, 0.81] for sin 54 or cos 36 |
| \(\sqrt{(\text{their } 4.9(\ldots))^2 - 4^2}\) or \(\sqrt{[8, 8.45]}\) or their 4.9(…) \(\times \sin 36\) or their 4.9(…) \(\times \cos 54\) | M1dep | oe |
| [2.9, 2.91] with M2 awarded | A1 | accept 3 with M2 awarded |
| Alt 1 SOHCAHTOA with TOA selected (eg circled) | M1 |
| [2.9, 2.91] with no working | M0M0A0 |
| [2.9, 2.91] from accurate/scale drawing only | M0M0A0 |
22 Use trigonometry to work out the value of \(x\). [3 marks]
Not drawn accurately
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – using sin 40 | ||
| sin chosen or used | M1 | |
| \(21 \times \sin 40\) | M1dep | accept \(21 \times [0.64, 0.643]\) |
| [13.49, 13.5] | A1 | |
| Alternative method 2 – using cos 50 | ||
| \(\cos (90 - 40)\) | M1 | |
| \(21 \times \cos (90 - 40)\) | M1dep | oe accept \(21 \times [0.64, 0.643]\) |
| [13.49, 13.5] | A1 | |
| Alternative method 3 – finds base then uses Pythagoras | ||
| \(21^2 - (21 \sin (90 - 40))^2\) | M1 | oe complete method except square root |
| \(\sqrt{21^2 - (21 \sin (90 - 40))^2}\) or \(\sqrt{[182.2, 182.22]}\) | M1dep | oe |
| [13.49, 13.5] | A1 | |
| Check diagram for working | |
| Allow correct use of sine rule to indicate sin | |
| Ignore rounding or truncating after the correct answer is seen eg answer 14 after 13.5 seen | M1M1A1 |
| \(\sin 40 \times 21\) | M2 |
| sin may be indicated by eg circling S in SOH CAH TOA | |
| Do not accept answers from full sized or scale drawing | |
| sin 50 used (unless using Alt 3) | M0 |
19
Not drawn accurately
Use trigonometry to work out the value of \(x\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\sin\) chosen or used | M1 | |
| \(31 \times \sin 24\) | M1dep | accept \(31 \times [0.4, 0.41]\) |
| [12.6, 12.61] | A1 | accept 13 if M2 awarded |
| Alternative method 2 | ||
| \(\cos (90 - 24)\) | M1 | |
| \(31 \times \cos (90 - 24)\) | M1dep | accept \(31 \times [0.4, 0.41]\) |
| [12.6, 12.61] | A1 | accept 13 if M2 awarded |
| Check diagram for working | |
| Allow correct use of sine rule to indicate \(\sin 24\) | |
| Ignore rounding or truncating after the correct answer is seen | |
| \(\sin 24 \times 31\) | M2 |
| Do not accept answers from full sized or scale drawing | |
| \(\sin\) may be indicated by eg circling S in SOH CAH TOA |
23 Use trigonometry to work out the size of angle \(w\). [3 marks]
Not drawn accurately
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| cos chosen or used | M1 | |
| \(\cos w = \dfrac{6.7}{8.3}\) or \(\cos^{-1} \dfrac{6.7}{8.3}\) | M1dep | any letter or symbol for \(w\) accept 0.807(…) or 0.81 for \(\dfrac{6.7}{8.3}\) |
| [35.9, 36.2] | A1 | |
| Alternative method 2 | ||
| \(\sin x = \dfrac{6.7}{8.3}\) or \(\sin^{-1} \dfrac{6.7}{8.3}\) or [53.8, 54.1] | M1 | any letter or symbol other than \(w\) accept 0.807(…) or 0.81 for \(\dfrac{6.7}{8.3}\) |
| \(90 -\) their [53.8, 54.1] | M1dep | |
| [35.9, 36.2] | A1 | |
| Alternative method 3 | ||
| \(\sqrt{8.3^2 - 6.7^2}\) or \(\sqrt{68.89 - 44.89}\) or \(\sqrt{24}\) or \(2\sqrt{6}\) or [4.89, 4.9] and \(\sin^{-1} \dfrac{\text{their } [4.89, 4.9]}{8.3}\) or \(\tan^{-1} \dfrac{\text{their } [4.89, 4.9]}{6.7}\) | M2 | full method to work out the missing length and use it correctly to work out the value of \(w\) any letter or symbol for \(w\) |
| [35.9, 36.2] | A1 | |
| Use of sine rule follows Alt method 2 | |
| \(\sin w = \dfrac{6.7}{8.3}\) without \(\sin^{-1} \dfrac{6.7}{8.3}\) or [53.8, 54.1] | M0 |
| \(\cos w = 0.807\) | M1M1 |
| \(\cos^{-1} w = \dfrac{6.7}{8.3}\) or \(\cos = \dfrac{6.7}{8.3}\) unless recovered | M1M0 |
29 Use trigonometry to work out the length \(x\).
Not drawn accurately
[2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\sin 72 = \dfrac{x}{8}\) or \(8 \times \sin 72\) or \(\cos(90 - 72) = \dfrac{x}{8}\) or \(8 \times \cos(90 - 72)\) or \(\dfrac{x}{\sin 72} = \dfrac{8}{\sin 90}\) or \(\dfrac{\sin 72}{x} = \dfrac{\sin 90}{8}\) | M1 | oe eg \(8\cos 72\) or 2.47… or 2.5 and \(\sqrt{8^2 - (8\cos 72)^2}\) |
| [7.6, 7.61] | A1 |
| If trigonometry and Pythagoras are used it must be a fully correct method that would lead to the correct value of \(x\) | |
| Accept \(\sin 72 \times 8\) | M1 |
| Accept opp or o for \(x\) \(\;\) eg \(\;\sin 72 = \dfrac{\text{opp}}{8}\) | M1 |
| \(\sin = \dfrac{x}{8}\) or \(\sin\theta = \dfrac{x}{8}\) (unless recovered) | M0 |
| Answer coming from scale drawing | M0A0 |
| Answer in range seen followed by 7 or 8 | M1A1 |
7 Use trigonometry to work out the length \(x\).
Not drawn accurately
[2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\sin 72 = \dfrac{x}{8}\) or \(8 \times \sin 72\) or \(\cos (90 - 72) = \dfrac{x}{8}\) or \(8 \times \cos (90 - 72)\) or \(\dfrac{x}{\sin 72} = \dfrac{8}{\sin 90}\) or \(\dfrac{\sin 72}{x} = \dfrac{\sin 90}{8}\) | M1 | oe eg \(8 \cos 72\) or 2.47… or 2.5 and \(\sqrt{8^2 - (8\cos 72)^2}\) |
| [7.6, 7.61] | A1 |
| If trigonometry and Pythagoras are used it must be a fully correct method that would lead to the correct value of \(x\) | |
| Accept \(\sin 72 \times 8\) | M1 |
| Accept opp or o for \(x\) \(\;\) eg \(\sin 72 = \dfrac{\text{opp}}{8}\) | M1 |
| \(\sin = \dfrac{x}{8}\) or \(\sin \theta = \dfrac{x}{8}\) (unless recovered) | M0 |
| Answer coming from scale drawing | M0A0 |
| Answer in range seen followed by 7 or 8 | M1A1 |
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