M151Q Precalculus · Unit 4: Trigonometric functions · on Final Exam
TRIG 3.1–3.3 Obtuse angles, Law of Sines, Law of Cosines
Study notes for Obtuse angles, Law of Sines, Law of Cosines in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Coordinate definitions extend the ratios to any angle: for a point $(x,y)$ at distance $r$ from the origin on the terminal side, $\sin\theta = y/r$, $\cos\theta = x/r$, $\tan\theta = y/x$. Obtuse angles have negative cosine and positive sine: $\sin(180^\circ - \theta) = \sin\theta$, $\cos(180^\circ - \theta) = -\cos\theta$.
- Law of Sines $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$: use with two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA, the ambiguous case: check for a second triangle with $180^\circ - B$).
- Law of Cosines $c^2 = a^2 + b^2 - 2ab\cos C$: use with two sides and the included angle (SAS) or three sides (SSS). It reduces to Pythagoras when $C = 90^\circ$.
- Area of any triangle: $\tfrac12 ab\sin C$.
Key formulas
Law of Sines
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$
Law of Cosines
$$c^2 = a^2 + b^2 - 2ab\cos C,\qquad \cos C = \frac{a^2 + b^2 - c^2}{2ab}$$
Area
$$\text{Area} = \tfrac12 ab\sin C$$
Worked example
In triangle $ABC$, $a = 8$, $b = 5$, $C = 60^\circ$. Find $c$ and angle $A$.
Show the solution
$c^2 = 64 + 25 - 80\cos60^\circ = 49 \Rightarrow c = 7$. Law of Sines: $\sin A = 8\sin60^\circ/7 \approx 0.990 \Rightarrow A \approx 81.8^\circ$ (since $a$ is the longest side, $A$ is the largest angle; acute here).
Common mistakes
- Using the Law of Sines with SSS or SAS (it needs a matched side–angle pair).
- Missing the second triangle in the ambiguous case.
- Sign error in the −2ab cos C term for obtuse angles (cos is negative, the term adds).
Exam tipCount the givens: two angles → Law of Sines; two sides with the included angle or three sides → Law of Cosines first.
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