Trigonometry Test #4 Review – Laws of Sines and Cosines, Vectors, and Trigonometric Equations

Trigonometric Equations and Solutions

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy a given equation within a specified interval, often [0, 2π) or [0°, 360°). Common strategies include factoring, using identities, and applying the unit circle.

• Quadratic Equations in Trigonometric Functions: Equations like 4sin2x - 3 = 0 can be solved by isolating the trigonometric function and using inverse functions.

• Using the Unit Circle: The unit circle helps identify all solutions for equations such as sin x = 1/2 or cos x = 0 within a given interval.

• Multiple Angle Equations: For equations like cos 3x = 1/2, solve for 3x first, then divide the solutions by 3 to find all x in the interval.

Example: Solve sin x = 1/2 for x in [0°, 360°): The solutions are x = 30°, 150° (or x = π/6, 5π/6 in radians).

Law of Sines and Law of Cosines

Law of Sines

The Law of Sines relates the sides and angles of any triangle, not just right triangles. It is especially useful for solving triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

• Formula:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

• Application: Use to find unknown sides or angles.

• Ambiguous Case (SSA): May yield 0, 1, or 2 possible triangles.

Example: Given sin 63°/a = sin 52°/7, solve for a:

$\frac{\sin 63^\circ}{a} = \frac{\sin 52^\circ}{7}$ $\Rightarrow a = \frac{7 \sin 63^\circ}{\sin 52^\circ}$

Law of Cosines

The Law of Cosines generalizes the Pythagorean Theorem for any triangle and is used when two sides and the included angle (SAS) or all three sides (SSS) are known.

• Formulas:

$a^2 = b^2 + c^2 - 2bc \cos A$ $b^2 = a^2 + c^2 - 2ac \cos B$ $c^2 = a^2 + b^2 - 2ab \cos C$

• Application: Use to find unknown sides or angles, especially when the Law of Sines is not applicable.

Example: Find angle B if a = 3.1, b = 7.3, c = 5.4:

$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$

Area of a Triangle

Area Formulas

The area of a triangle can be found using trigonometric relationships when two sides and the included angle are known, or using Heron's formula when all three sides are known.

• Trigonometric Area Formula:

$\text{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} ac \sin B = \frac{1}{2} bc \sin A$

• Heron's Formula:

$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{1}{2}(a+b+c)$

Example: Find the area if a = 8 in, b = 12 in, c = 16 in:

$s = \frac{1}{2}(8+12+16) = 18$ $\text{Area} = \sqrt{18(18-8)(18-12)(18-16)}$

Ambiguous Case (SSA) and Number of Triangles

Determining the Number of Possible Triangles

When given two sides and a non-included angle (SSA), the Law of Sines may yield zero, one, or two possible triangles. This is known as the ambiguous case.

• Zero Triangles: No solution if the side opposite the given angle is too short.

• One Triangle: Unique solution if the side is long enough or the angle is right.

• Two Triangles: Two possible solutions if the side is long enough to form two different triangles.

Example: Given a = 20, b = 12, B = 25°, determine the number of triangles possible using the Law of Sines.

Vectors in Trigonometry

Vector Components

A vector in the plane can be described by its magnitude and direction, or by its horizontal (x) and vertical (y) components.

• Component Formulas:

$\vec{v} = \langle v_x, v_y \rangle$ $v_x = |\vec{v}| \cos \theta$ $v_y = |\vec{v}| \sin \theta$

Example: For $|\vec{v}| = 6$, $\theta = 203^\circ$:

$v_x = 6 \cos 203^\circ$ v_y = 6 \sin 203^\circ$

Dot Product and Angle Between Vectors

The dot product of two vectors is a scalar value that can be used to find the angle between them.

• Dot Product Formula:

$\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2$

• Angle Between Vectors:

$\cos \theta = \frac{\vec{u} \cdot \vec{v}}{||\vec{u}||\ ||\vec{v}||}$

Example: For $\vec{u} = \langle 4, -2 \rangle$, $\vec{v} = \langle -1, 6 \rangle$:

$\vec{u} \cdot \vec{v} = 4 \times (-1) + (-2) \times 6 = -4 - 12 = -16$

Orthogonality of Vectors

Two vectors are orthogonal (perpendicular) if their dot product is zero.

• Test for Orthogonality: If $\vec{u} \cdot \vec{v} = 0$, then $\vec{u}$ and $\vec{v}$ are orthogonal.

Example: Are $\langle -4, 6 \rangle$ and $\langle 3, -2 \rangle$ orthogonal?

$(-4) \times 3 + 6 \times (-2) = -12 - 12 = -24 \neq 0$ So, they are not orthogonal.

Summary Table: Key Trigonometric Laws and Formulas

Law/Formula	Equation	When to Use
Law of Sines	$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$	ASA, AAS, SSA (ambiguous case)
Law of Cosines	$a^2 = b^2 + c^2 - 2bc \cos A$	SAS, SSS
Area (trig)	$\frac{1}{2} ab \sin C$	Two sides and included angle
Area (Heron's)	$\sqrt{s(s-a)(s-b)(s-c)}$	All three sides known
Dot Product	$\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2$	Angle between vectors, orthogonality
Vector Components	$v_x = |\vec{v}| \cos \theta$, $v_y = |\vec{v}| \sin \theta$	Magnitude and direction to components

Additional info:

• All equations are provided in LaTeX format for clarity and further study.

• The unit circle diagram is included to reinforce the connection between angle measures and trigonometric values.