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