Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
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Slope of a Line
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Percent of Change
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Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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Solving Right Triangles

Example 1

Solve the right triangle sketched below.


We are given A = 27º and b = 412 m

To solve this triangle, we need to compute values for B, a, and c.

When one of the acute angles is given, the easiest way to compute the other is by subtraction from 90º:

A + B = 90º (for a right triangle)

So, 27º + B = 90º giving B = 90º - 27º = 63º.

Then, for side a, we have or which gives a = (412 m)(tan 27º) ≈ 209.92 m.

Finally, for side c, or giving

Thus, the required solution is: B = 63º, a ≈ 209.92 m, c ≈ 462.40 m


Note the strategy employed in the previous example. To calculate ‘a’, we looked for a trigonometric ratio that involved A and b (our given quantities) and ‘a’ (the unknown we wished to determine). Similarly, to determine c, we looked for a trigonometric ratio involving A, b, and c. In both of these cases this meant that the definition of the selected trigonometric ratio would amount to an equation with one unknown, which was then easily solved.


Example 2

Solve the right triangle shown in the figure below.


The solution here is that there is no solution. Knowing all three angles of a right triangle is not enough information to be able to calculate the lengths of any of the three sides. In fact, there is an infinite number of right triangles that have these three angles – each of a different size.

So, we cannot obtain a unique solution for this triangle.


Example 3

Solve the right triangle with c = 26.8 cm and B = 37.56º.


It is probably best to start by making a sketch, as is done to the right.

In addition to the right angle, of course, the known parts of the triangle, B and c, are given in the statement of the problem. So, we need to determine the values of A, a, and b.

First, A + B = 90º so A + 37.56º = 90º giving A = 90º - 37.56º =52.44º

Then, gives or

b = (26.8 cm)(sin 37.56º) ≈ 16.34 cm

Finally, using or gives us

a = (26.8 cm)(cos 37.56º) ≈ 21.24 cm

Thus, the required solution is

A = 52.44º, a ≈ 21.24 cm, b ≈ 16.34 cm.



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