Topics › KS4 › Geometry & measures · Foundation & Higher

Right-angled trigonometry

1. trig-01 Present from here

   In triangle $ABC$, angle $B = 90^\circ$, angle $A = 35^\circ$ and $AC = 12\text{ cm}$. Work out the length of $BC$, to 1 decimal place.

   A Option A: $9.8\text{ cm}$

   B Option B: $20.9\text{ cm}$

   C Option C: $6.9\text{ cm}$

   D Option D: $8.4\text{ cm}$
   Answer and misconceptions

   A Option A: $9.8\text{ cm}$ ✗

   Uses cosine for the side opposite the angleWorks out $12 \times \cos 35^\circ$; $BC$ is opposite angle $A$ and $AC$ is the hypotenuse, so use $\sin$: $12 \times \sin 35^\circ = 6.9$.

   B Option B: $20.9\text{ cm}$ ✗

   Divides by the ratio instead of multiplyingWorks out $12 \div \sin 35^\circ$; from $\sin 35^\circ = \frac{BC}{12}$, multiply both sides by $12$.

   C Option C: $6.9\text{ cm}$ ✓ Correct

   D Option D: $8.4\text{ cm}$ ✗

   Uses tangent with the hypotenuseWorks out $12 \times \tan 35^\circ$; tangent links the opposite and adjacent sides, not the hypotenuse.
2. trig-02 Present from here

   In triangle $PQR$, angle $Q = 90^\circ$, $PQ = 8\text{ cm}$ and $QR = 5\text{ cm}$. Work out angle $P$, to 1 decimal place.

   A Option A: $32.0^\circ$

   B Option B: $58.0^\circ$

   C Option C: $38.7^\circ$

   D Option D: $51.3^\circ$
   Answer and misconceptions

   A Option A: $32.0^\circ$ ✓ Correct

   B Option B: $58.0^\circ$ ✗

   Swaps the opposite and adjacent sidesWorks out $\tan^{-1}\left(\frac{8}{5}\right)$; $QR$ is opposite angle $P$, so $\tan P = \frac{5}{8}$. $58.0^\circ$ is angle $R$.

   C Option C: $38.7^\circ$ ✗

   Treats the longest given side as the hypotenuseUses $\sin P = \frac{5}{8}$; the hypotenuse is $PR$, opposite the right angle, which is not given.

   D Option D: $51.3^\circ$ ✗

   Uses cosine for the opposite and adjacent sidesWorks out $\cos^{-1}\left(\frac{5}{8}\right)$; cosine links adjacent and hypotenuse. Opposite and adjacent need tan: $\tan^{-1}\left(\frac{5}{8}\right) = 32.0^\circ$.
3. trig-03 Present from here

   Without a calculator, what is the exact value of $\tan 60^\circ$?

   A Option A: $\frac{1}{\sqrt{3}}$

   B Option B: $\frac{\sqrt{3}}{2}$

   C Option C: $\frac{1}{2}$

   D Option D: $\sqrt{3}$
   Answer and misconceptions

   A Option A: $\frac{1}{\sqrt{3}}$ ✗

   Divides cos by sin instead of sin by cosWorks out $\cos 60^\circ \div \sin 60^\circ = \frac{1}{\sqrt{3}}$, which is $\tan 30^\circ$; $\tan 60^\circ = \frac{\sqrt{3}}{2} \div \frac{1}{2} = \sqrt{3}$.

   B Option B: $\frac{\sqrt{3}}{2}$ ✗

   Gives $\sin 60^\circ$ instead of $\tan 60^\circ$$\frac{\sqrt{3}}{2}$ is $\sin 60^\circ$; tan is sin divided by cos.

   C Option C: $\frac{1}{2}$ ✗

   Gives $\cos 60^\circ$ instead of $\tan 60^\circ$$\frac{1}{2}$ is $\cos 60^\circ$ (and $\sin 30^\circ$), not $\tan 60^\circ$.

   D Option D: $\sqrt{3}$ ✓ Correct
4. trig-04 Present from here

   In triangle $XYZ$, angle $Y = 90^\circ$, angle $X = 40^\circ$ and $XY = 7\text{ cm}$. Work out the length of the hypotenuse $XZ$, to 1 decimal place.

   A Option A: $5.4\text{ cm}$

   B Option B: $9.1\text{ cm}$

   C Option C: $10.9\text{ cm}$

   D Option D: $8.3\text{ cm}$
   Answer and misconceptions

   A Option A: $5.4\text{ cm}$ ✗

   Multiplies when the unknown is on the bottomFrom $\cos 40^\circ = \frac{7}{XZ}$, works out $7 \times \cos 40^\circ$; the unknown is the denominator, so $XZ = 7 \div \cos 40^\circ$. A hypotenuse must be the longest side.

   B Option B: $9.1\text{ cm}$ ✓ Correct

   C Option C: $10.9\text{ cm}$ ✗

   Uses sine for the side next to the angleWorks out $7 \div \sin 40^\circ$; $XY$ touches angle $X$, so it is the adjacent side and the ratio is cosine.

   D Option D: $8.3\text{ cm}$ ✗

   Uses tangent when the hypotenuse is involvedWorks out $7 \div \tan 40^\circ$; tangent links opposite and adjacent, so it cannot find the hypotenuse.
5. trig-05 Present from here

   In triangle $DEF$, angle $E = 90^\circ$, $DF = 10\text{ cm}$ and $EF = 7\text{ cm}$. Work out angle $D$, to 1 decimal place.

   A Option A: $44.4^\circ$

   B Option B: $45.6^\circ$

   C Option C: $35.0^\circ$

   D Option D: $0.8^\circ$
   Answer and misconceptions

   A Option A: $44.4^\circ$ ✓ Correct

   B Option B: $45.6^\circ$ ✗

   Takes the opposite side as the adjacent sideUses $\cos D = \frac{7}{10}$; $EF$ does not touch angle $D$, so it is opposite: $\sin D = \frac{7}{10}$. $45.6^\circ$ is angle $F$.

   C Option C: $35.0^\circ$ ✗

   Uses tangent with the hypotenuseUses $\tan D = \frac{7}{10}$; $10\text{ cm}$ is the hypotenuse, so the ratio is sine.

   D Option D: $0.8^\circ$ ✗

   Calculator set to radians, not degrees$\sin^{-1}\left(\frac{7}{10}\right)$ in radians mode gives $0.775\ldots$; in degrees mode it gives $44.4^\circ$.