Solving Oblique Triangles

Triangle Properties and Classification

Oblique triangles are triangles that do not contain a right angle. To solve such triangles, at least three of the six parts (three sides and three angles) must be known, with at least one part being a side. Key properties include:

• Sum of angles: The sum of the measures of the three angles is always 180°.

• Side length relationships: The sum of the lengths of any two sides must be greater than the third side.

• Opposite relationships: The longest side is opposite the largest angle; the shortest side is opposite the smallest angle.

Strategies for Solving Triangles

The method used to solve a triangle depends on the combination of known sides and angles:

Law of Sines

Definition and Formula

The Law of Sines relates the sides and angles of an oblique triangle. It states that the ratio of the sine of an angle to the length of the side opposite that angle is constant for all three pairings:

• Law of Sines:

Or equivalently:

When to Use the Law of Sines

• Given one side and two angles (ASA or AAS).

• Given two sides and a non-included angle (SSA, ambiguous case).

Ambiguous Case (SSA)

When two sides and a non-included angle are given, the number of possible triangles depends on the relationship between the sides and the height:

• No triangle if

• One right triangle if

• One triangle if

• Two triangles if

Additional info: The height is calculated as .

Area of Triangles

Area Formulas for Oblique Triangles

The area of a triangle can be calculated using the base and height, or using the Law of Sines:

• General formula:

• Using Law of Sines:

Application Problems: Bearings and Directions

Bearings

Bearings describe direction relative to due North, measured clockwise. For example, a bearing of 90° is due East.

Degrees and Direction from North or South

Directions can also be given as degrees from North or South, such as N38°E or S15°E.

Parallel Lines and Traversals

When solving application problems, knowledge of angles formed by a transversal intersecting parallel lines is useful. Key angle relationships include corresponding, alternate interior, alternate exterior, and vertical angles.

Law of Cosines

Definition and Formula

The Law of Cosines is used when three sides (SSS) or two sides and the included angle (SAS) are known. It generalizes the Pythagorean Theorem for oblique triangles:

Solving SSS and SAS Triangles

• For SSS: Find the largest angle using Law of Cosines, then use Law of Sines for smaller angles.

• For SAS: Find the unknown side using Law of Cosines, then use Law of Sines for remaining angles.

Heron's Area Formula

For triangles with all three sides known:

where

Length of a Chord in a Circle

The length of a chord in a circle with radius and central angle :

Vectors

Definition and Properties

Vectors are directed line segments with both magnitude and direction. Examples include velocity, force, and displacement. The magnitude is denoted .

Scalar Multiplication

• Multiplying a vector by a scalar changes its magnitude and possibly its direction.

Resultant Vector

The sum of two vectors is called the resultant vector, found by the parallelogram method.

Components and Position Vector

Vectors can be broken into horizontal and vertical components. The position vector in component form is .

Magnitude and Direction Angle

• Magnitude:

• Direction angle (counterclockwise from x-axis):

Vector Operations

• Scalar Product:

• Vector Sum:

• Vector Difference:

• Dot Product:

Angle Between Two Vectors

The angle between vectors and :

• If , vectors are perpendicular.

• If , vectors are parallel.

Unit Vectors

Unit vectors and are used to express any vector as .

Polar Equations

Polar Coordinates

Polar coordinates describe a point by its distance from the pole (origin) and angle from the polar axis (positive x-axis).

• Plot angle first, then distance .

• can be positive or negative.

• Multiple representations for the same point are possible.

Conversion Between Rectangular and Polar Coordinates

• From rectangular to polar: ,

• From polar to rectangular: ,

Parametric Equations

Definition and Graphing

Parametric equations express and as functions of a parameter . The graph is produced by plotting for values of within a specified interval.

Eliminating the Parameter

To convert parametric equations to a rectangular equation, solve for in one equation and substitute into the other.

Projectile Motion Application

For a projectile thrown at angle with initial velocity and height :

• Horizontal distance:

• Vertical height:

Additional info: The coefficient -16 is for feet/sec2; use -4.9 for meters/sec2.