Chapter 3: Laws of Sines and Cosines
Chapter 3 Summary and Review
Key Concepts
- We put an angle [latex]\theta[/latex] in standard position by placing its vertex at the origin and the initial side on the positive [latex]x[/latex]-axis.
- Coordinate Definitions of the Trigonometric Ratios.
[latex]\cos \theta = \dfrac{x}{r}[/latex]
[latex]\sin \theta = \dfrac{y}{r}[/latex]
[latex]\tan \theta = \dfrac{y}{x}[/latex] - Trigonometric Ratios for Supplementary Angles.
[latex]\cos(180° - \theta) = -\cos \theta[/latex]
[latex]\sin(180° - \theta)= \sin \theta[/latex]
[latex]\tan(180° - \theta) = -\tan \theta[/latex] - There are always two [latex]supplementary[/latex] angles between [latex]0°[/latex] and [latex]180°[/latex] that have the same sine. Your calculator will only tell you one of them.
- Area of a Triangle.
If a triangle has sides of length [latex]a[/latex] and [latex]b{,}[/latex] and the angle between those two sides is [latex]\theta{,}[/latex] then the area of the triangle is given by
[latex]A = \dfrac{1}{2} ab \sin \theta[/latex]
- Law of Sines.
If the angles of a triangle are [latex]A, B{,}[/latex] and [latex]C{,}[/latex] and the opposite sides are, respectively, [latex]a, b,[/latex] and [latex]c{,}[/latex] then
[latex]\dfrac {\sin A}{a} = \dfrac {\sin B}{b} = \dfrac {\sin C}{c}[/latex]
or equivalently,
[latex]\dfrac {a}{\sin A} = \dfrac {b}{\sin B} = \dfrac {c}{\sin C}[/latex]
- We can use the law of sines to find an unknown side in an oblique triangle. We must know the angle opposite the unknown side and another side-angle pair.
- We can also use the law of sines to find an unknown angle of a triangle. We must know two sides of the triangle and the angle opposite one of them.
- Remember that there are two angles with a given \sine. When using the law of sines, we must check whether both angles result in possible triangles.
- We use minutes and seconds to measure very small angles. Fractions of a °.
One minute: [latex]~~~~~~1^{\prime} = \dfrac{1°}{60}[/latex]
One second: [latex]~~~~~~1^{\prime \prime} = \dfrac{1^{\prime}}{60} = \dfrac{1°}{3600}[/latex] - You can remember the trig values for the special angles if you memorize two triangles:
- For the trigonometric ratios of most angles, your calculator gives approximations, not exact values.
- The law of sines is not helpful when we know two sides of the triangle and the included angle. In this case we need the law of cosines.
- Law of Cosines.
If the angles of a triangle are [latex]A, B{,}[/latex] and [latex]C{,}[/latex] and the opposite sides are, respectively, [latex]a, b,[/latex] and [latex]c{,}[/latex] then
[latex]a^{2} = b^{2} + c^{2} - 2bc \cos A[/latex]
[latex]b^{2} = a^{2} + c^{2} - 2ac \cos B[/latex]
[latex]c^{2} = a^{2} + b^{2} - 2ab \cos C[/latex] - We can also use the law of cosines to find an angle when we know all three sides of a triangle.
- We can use the law of cosines to solve the ambiguous case.
- How to Solve an Oblique Triangle.
If we know: We can use: 1. One side and two angles (SAA) 1. Law of sines to find another side 2. Two sides and the angle opposite
one of them (SSA, the ambiguous
case)2. Law of sines to find another angle,
or law of cosines to find another
side3. Two sides and the included angle
(SAS)3. Law of cosines to find the third
side4. Three sides (SSS) 4. Law of cosines to find an angle