Precalculus Practice: Trigonometry, Angles, and Polygons

Solving Right Triangles and Angle Properties

Understanding the Sine of an Angle

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Since side lengths cannot be negative, the sine of an angle must be less than one.

• Key Point: For any angle in a right triangle, $0 \leq \sin(\theta) \leq 1$.

• Example: If the opposite side is 3 and the hypotenuse is 5, $\sin(\theta) = \frac{3}{5} = 0.6$.

Solving Right Triangles

Using Trigonometric Ratios

To solve a right triangle, use the definitions of sine, cosine, and tangent to find unknown sides or angles.

• Given: $a = 10^2$, $b = 20$, $c = 32^2$

• Formula: Pythagorean Theorem: $a^2 + b^2 = c^2$

• Example: If $a = 10$, $b = 20$, then $c = \sqrt{10^2 + 20^2} = \sqrt{100 + 400} = \sqrt{500} \approx 22.36$

Solving for an Angle

• Given: $c = h$, $a = 40^2$, $b = 0.32^2$

• Formula: Use inverse trigonometric functions to find angles: $\theta = \arccos\left(\frac{a}{c}\right)$ or $\theta = \arcsin\left(\frac{b}{c}\right)$

Trigonometric Applications

Angles of Elevation and Depression

Angles of elevation and depression are used to describe the angle above or below the horizontal line of sight.

• Key Point: The angle of elevation is measured upward from the horizontal, while the angle of depression is measured downward.

• Example: If a pilot sees a plane at an angle of depression of $25^\circ$ and another at $13^\circ$, the vertical distance between the planes can be found using trigonometric ratios and the altitude.

Solving Word Problems with Trigonometry

• Example: A plane is flying directly above a line connecting two towns. The distance between the towns is 8 miles, and the angle of elevation from each town to the plane is given. Use the Law of Sines or Law of Cosines to find the height of the plane.

• Formula (Law of Sines): $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Polygons and Regular Figures

Interior Angles and Diagonals

The sum of the interior angles of a polygon with $n$ sides is given by:

• $\text{Sum of interior angles} = (n-2) \times 180^\circ$

• Example: For a regular heptagon ($n=7$), each interior angle is $\frac{(7-2) \times 180^\circ}{7} = \frac{900^\circ}{7} \approx 128.57^\circ$

• Diagonals: The number of diagonals in a polygon is $\frac{n(n-3)}{2}$

Finding the Length of a Diagonal

• Given: A regular heptagon with sides of length 6 inches.

• Formula: Use the Law of Cosines to find the length of a diagonal.

• $d^2 = 2a^2 - 2a^2 \cos\left(\frac{5\pi}{7}\right)$, where $a$ is the side length.

Patterns and Rotational Symmetry

Rotational Symmetry of Polygons

Regular polygons have rotational symmetry. The number of unique images formed when a polygon is rotated is equal to the number of sides.

• Example: A regular polygon with 4, 6, 8, 10, 12, 20, and 24 sides will have 4, 6, 8, 10, 12, 20, and 24 unique images, respectively, at each increment of rotation.