A triangle ABC is such that AB = 6 cm,angle A = 12° and BC = 2.5 cm. Find the two possible values of angle C.
There has never been an exam question like this, but the course specification states explicitly that "related angles could be combined with sine and cosine rules" so you need to be able to handle this.
This question needs an understanding of the so-called "ambiguous case" of the sine rule. There are actually two different ways to draw a triangle with AB = 6 cm,angle A = 12° and BC = 2.5 cm – try it and see! So there are two different possible angles for C: one acute and one obtuse.
However, we'll worry about that at the end. We start in the normal way:
So that's one possible value of C. But from our knowledge of the quadrant diagram from studying trig equations, there is a related angle in the 2nd quadrant. It is 180 – 29.9 ° = 150.1°.
Therefore, angle C = 29.9° or 150.1° (correct to 1 decimal place).
Frank and Sean are standing 54 metres apart on a horizontal football pitch. Frank is at F and Sean is at S. They are each looking at a stationary drone D in the air above the pitch.
The angle of inclination of D from F is 38°.
The angle of inclination of D from S is 49°.
Calculate the height of the drone above the ground, correct to 3 significant figures.
A question very similar to this appeared in 2019 Paper 2. It caught a lot of candidates out, because they weren't expecting to have to use SOH CAH TOA (a National 4 topic) as well as the sine rule.
The method here is to find either FD or SD, then draw the vertical height below D and use right-angled trigonometry to find the height.
We will find FD. You might like to try this yourself, but finding SD instead.
First, we need to find angle D, because it's opposite the 54 metres:
In this diagram:
• angle ABD = 75°
• angle BDC = 37°
• BC = 20 centimetres
Calculate the length of DC.
This 3-mark question needs us to calculate the angle CBD before using the sine rule.
The question really should have told us that ABC is a straight line, but as answering it would be impossible without knowing this, we can safely assume it. So \(\angle\)CBD \(=180-75=105^\circ\small.\)
Katy and Mona are looking up at a hot-air balloon.
In the diagram below, K, M and B represent the positions of Katy, Mona and the balloon respectively.
• The angle of elevation of the balloon from Katy is 52°
• The angle of elevation of the balloon from Mona is 34°
• Katy and Mona are 350 metres apart on level ground.
Calculate the height of the hot-air balloon above the ground.
First, we need to use the sine rule to find the length of either KB or BM. It doesn't matter which. We will find KB.
Note that angle KBM = 180 – 52 – 34 = 94°. We need this because it is opposite the given side.
Click here to study the sine rule notes on National5.com.
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