Trig Ratios and Laws Practice

These questions follow Special Angles, The Sine Law, and The Cosine Law. The first group is calculator-free; after that, half the skill is choosing the tool before touching a button. Round to one decimal place where rounding is needed.

Exact values and the full circle

1. State exact values: $\sin 60°$, $\tan 45°$, $\cos 30°$.
2. Evaluate exactly: ${\sin }^{2}45°+{\cos }^{2}45°$.
3. Find both angles between $0°$ and $360°$ with $\sin \theta =\frac{1}{2}$.
4. Find both angles between $0°$ and $360°$ with $\tan \theta =1$.

Answer 1

From the two rebuildable triangles: $\sin 60°=\frac{\sqrt{3}}{2}$, $\tan 45°=1$, $\cos 30°=\frac{\sqrt{3}}{2}$. If you drew the triangles rather than recalling digits, the answers came with receipts.

Answer 2

${\left(\frac{\sqrt{2}}{2}\right)}^2+{\left(\frac{\sqrt{2}}{2}\right)}^2=\frac{1}{2}+\frac{1}{2}=1$. Not a coincidence: on the unit circle, $\sin \theta$ and $\cos \theta$ are the legs of a right triangle whose hypotenuse is 1, so this sum is 1 at every angle.

Answer 3

$\theta =30°$ or $\theta =150°$. The rotating arm reaches height $\frac{1}{2}$ once rising, once returning — the second angle is $180°-30°$.

Answer 4

$\theta =45°$ or $\theta =225°$. Tangent is positive where sine and cosine share a sign — first and third quadrants — so the partner angle is $180°+45°$, not $180°-45°$.

Sine law — including the trap

5. In $△ABC$, $\angle A=44°$, $\angle B=71°$, and $a=12$ cm. Find $b$.
6. In $△ABC$, $\angle A=30°$, $a=6$ cm, and $b=10$ cm. Find all possible measures of $\angle B$.

Answer 5

A known side faces a known angle, so the sine law applies: $b=\frac{12\sin 71°}{\sin 44°}\approx 16.3$ cm. Check before believing: $71°>44°$, so $b$ had to exceed 12 cm. ✓

Answer 6

$\sin B=\frac{10\sin 30°}{6}=\frac{5}{6}$, so $B\approx 56.4°$ — or its supplement, $B\approx 123.6°$. Test both against the angle budget: $30°+56.4°<180°$ and $30°+123.6°<180°$, so both triangles exist and both answers stand. Reporting only the acute one is the classic miss; the question’s word “all” was the warning.

Cosine law and mixed

7. In $△ABC$, $b=8$ cm, $c=11$ cm, and the contained angle $\angle A=52°$. Find $a$.
8. A triangle has sides 6 m, 7 m, and 10 m. Find its largest angle.
9. Challenge. From point $P$, the angle of elevation to the top of a tower is $28°$. From point $Q$, which is 50 m closer along flat ground, it is $40°$. Find the height of the tower.

Answer 7

No known side faces a known angle — cosine law: $a^2=8^2+11^2-2(8)(11)\cos 52°\approx 76.6$, so $a\approx 8.8$ cm.

Answer 8

The largest angle faces the 10 m side: $\cos \theta =\frac{6^2+7^2-10^2}{2(6)(7)}=-\frac{15}{84}$, so $\theta \approx 100.3°$. The negative cosine announced an obtuse angle before the calculator finished — that is the check.

Answer 9

In $△PQT$ ($T$ the tower top): the angle at $Q$ is $180°-40°=140°$, leaving $12°$ at $T$. Sine law gives the slant side $QT=\frac{50\sin 28°}{\sin 12°}\approx 112.9$ m. Then the right triangle under $T$ gives height $h=QT\sin 40°\approx 72.6$ m. Two triangles sharing an edge — solve one, carry the edge to the other.