Question

k/u app thk com
16 12 10 6
communication in all questions must include: (c:3)

• enough steps shown to clearly demonstrate thinking
• solutions that are neat and easy to follow
• proper use of mathematical symbols
• equal signs aligned
• units used as required
• concluding statements for all word problems
• correct rounding.

knowledge/understanding: 16 marks
part a: multiple choice. circle the letter, which corresponds to the correct answer. 5 marks tot
in \\( \triangle a b c \\) on the right, what is \\( \sin a \\) ?
(a) \\( \frac{a}{c} \\) b) \\( \frac{b}{c} \\) c) \\( \frac{a}{b} \\) d) \\( \frac{b}{a} \\)
in the triangle on the right, what is the value of \\( \tan a \\) ?
a) \\( \frac{8}{17} \\) b) \\( \frac{15}{8} \\) c) \\( \frac{17}{15} \\) d) \\( \frac{8}{15} \\)
solving a triangle means:
a) finding the length of an unknown side.
b) finding the measure of an unknown angle.
c) determining the measures of any missing angles and the lengths of any missing sides.
d) determining whether a triangle is equilateral, isosceles or scalene.
which tool or strategy could you use to find the third side of this triangle?
a) the sine law
b) the cosine law
c) the pythagorean theorem
d) there is not enough information to find the third side.
you are told that \\( \cos a \\) is a negative number. what can you say about angle \\( a \\) ?
a) \\( a \\) is an acute angle c) \\( a \\) is an obtuse angle
b) \\( a \\) could be either acute or obtuse d) \\( a \\) is neither acute nor obtuse

Explanation:

Step1: Recall SOH - CAH - TOA

SOH - CAH - TOA is a mnemonic for trigonometric ratios. $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle A$ in $\triangle ABC$, the side opposite to $\angle A$ is $a$ and the hypotenuse is $c$. So, $\sin A=\frac{a}{c}$.

Step2: Recall $\tan\theta$ formula

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In the second triangle, for $\angle A$, the side opposite is $8$ and the side adjacent is $15$. So, $\tan A = \frac{8}{15}$.

Step3: Define solving a triangle

Solving a triangle means finding all unknown angles and sides. So, it is determining the measures of any missing angles and the lengths of any missing sides.

Step4: Choose the law for side - finding

When two sides and the included angle are known (here two sides \(18m\) and \(20m\) and included angle \(100^{\circ}\)), we use the cosine law. The cosine law is \(a^{2}=b^{2}+c^{2}-2bc\cos A\).

Step5: Analyze the sign of cosine

We know that \(\cos\theta> 0\) for \(0^{\circ}<\theta<90^{\circ}\) (acute angles) and \(\cos\theta<0\) for \(90^{\circ}<\theta<180^{\circ}\) (obtuse angles). If \(\cos A<0\), then \(A\) is an obtuse angle.

Answer:

1. a) \(\frac{a}{c}\)
2. d) \(\frac{8}{15}\)
3. c) determining the measures of any missing angles and the lengths of any missing sides
4. b) the cosine law
5. c) \(A\) is an obtuse angle