The is a powerful tool for solving . It connects the ratios of sines of angles to opposite side lengths, allowing us to find missing sides or angles when given partial information about a triangle.
This law has real-world applications in navigation, construction, and surveying. It can be used to calculate distances across bodies of water, determine heights of tall structures, and even design roof trusses. The Law of Sines expands our problem-solving toolkit beyond right triangles.
Law of Sines
Law of Sines for non-right triangles
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States the ratio of the sine of an angle to the length of the side opposite that angle is constant in any triangle ABC: asinA=bsinB=csinC
A, B, C represent the angles of the triangle
a, b, c represent the lengths of the sides opposite to angles A, B, C respectively (side a is opposite angle A)
Requires knowing at least one of the following to solve a non-right triangle:
Measure of one angle and its opposite side length (AAS or ASA)
Measures of two angles (AAA)
Lengths of two sides and measure of a non-included angle (SSA)
Steps to solve when given AAS or ASA:
Write out the Law of Sines formula
Plug in the known values into the formula
Solve the equation for the unknown side length or angle measure
Find the remaining unknown side or angle using the Law of Sines again
Verify results by checking the sum of all angles equals 180°
Steps to solve when given AAA:
Calculate the measure of the third angle using the triangle angle sum of 180°
Assign a variable like x to represent the length of any chosen side
Write out the Law of Sines formula using x for the chosen side
Solve the equation for x to determine the length of the chosen side
Use the Law of Sines again to find the lengths of the other two sides
When given SSA, there may be zero, one, or two possible solutions ()
No solution if angle is acute and opposite side shorter than known side
One solution (right triangle) if angle is acute and opposite side equals the product of known side and sine of angle
Two solutions if angle is acute and opposite side longer than product of known side and sine of angle
One solution if the given angle is obtuse
The Law of Sines is based on trigonometric ratios and can be derived using the unit circle
Area calculation with sine functions
Calculates area of a triangle using sine function and lengths of any two sides
Formula is Area=21absinC where:
a and b are the lengths of any two sides
C is the angle measure between the two chosen sides
Steps to find area of a non-right triangle using sine function:
Select any two sides of the triangle
Determine the angle measure between the selected sides
Plug the values into the area formula
Solve the equation for the area
Real-world applications of Law of Sines
Identify given information like angle measures or side lengths
Determine the appropriate solving method based on given info (AAS, ASA, AAA, SSA)
Sketch a diagram representing the problem, label known and unknown values
Apply Law of Sines to solve for unknowns, following steps for the scenario
Interpret the results in the context of the real-world application
Calculate triangle area with sine function if needed for the problem
Examples:
Calculating distance across a lake between two points on opposite shores
Determining height of a tree, building, or mountain using angle of elevation and distance from base
Navigating a boat between two islands given bearings and a distance
Designing a ramp or roof truss based on a required angle and span
Additional Concepts Related to Law of Sines
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side
Supplementary angles: When two angles add up to 180°, they are supplementary. This concept is useful when working with exterior angles of triangles
Periodic functions: Sine functions are periodic, repeating their values at regular intervals, which is important when considering multiple solutions in the ambiguous case
Key Terms to Review (7)
AAS (angle-angle-side): AAS (angle-angle-side) is a congruence criterion for triangles, stating that two triangles are congruent if two angles and the non-included side of one triangle are equal to the corresponding parts of another triangle. This method is used to determine the uniqueness of a triangle.
Altitude: Altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. It helps in determining the height of the triangle for various calculations.
Ambiguous case: The ambiguous case occurs when solving for a triangle using the Law of Sines and given two sides and a non-included angle (SSA). This situation can result in zero, one, or two possible triangles.
Law of Sines: The Law of Sines is a trigonometric equation that relates the lengths of the sides of a triangle to the sines of its angles. It states that for any triangle, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, where $a$, $b$, and $c$ are the lengths of the sides opposite angles $A$, $B$, and $C$ respectively.
Non-right triangles: Non-right triangles are triangles that do not have a 90-degree angle. Solving these triangles often involves using trigonometric laws such as the Law of Sines or the Law of Cosines.
Oblique triangle: An oblique triangle is a triangle that does not contain a right angle. It can be either an acute triangle, where all angles are less than 90 degrees, or an obtuse triangle, where one angle is greater than 90 degrees.
SSA (side-side-angle): SSA (Side-Side-Angle) is a condition in which two sides and a non-included angle of a triangle are known. It is also known as the ambiguous case because it can result in zero, one, or two possible triangles.