Navigation and Trigonometric Functions Summary

Types of Angles
  • Acute Angles: Angles that are greater than $0$ degrees and less than $90$ degrees.
  • Obtuse Angles: Angles that are greater than $90$ degrees.
  • When navigating, small angles can lead to widely divergent paths over large distances.
  • Degrees: A circle is divided into degrees, where:
    • 1 degree = 60 minutes (')
    • 1 minute = 60 seconds (")
  • Example: A precise measurement can be given as 35 degrees, 14 minutes, and 23 seconds, which can also be represented in decimal form as 35.5 degrees.
Converting Between Degrees and Radians
  • A degree can be converted into radians using:
    1extdegree=π180 radians1 ext{ degree} = \frac{\pi}{180} \text{ radians}
  • Conversely:
    1extradian=180π degrees1 ext{ radian} = \frac{180}{\pi} \text{ degrees}
Circle Measurements
  • The circumference of a circle can be described using radians:
    • C=2πrC = 2 \pi r
  • To convert from degrees to radians, multiply the number of degrees by π180\frac{\pi}{180}.
  • To convert from radians to degrees, multiply the number of radians by 180π\frac{180}{\pi}.
Trigonometric Functions
  • Trigonometric ratios arise in right triangles and include:
    • Sine ($\sin$): ratio of the length of the opposite side to the hypotenuse.
    • Cosine ($\cos$): ratio of the length of the adjacent side to the hypotenuse.
    • Tangent ($\tan$): ratio of the length of the opposite side to the adjacent side.
    • Cosecant ($\csc$): reciprocal of sine.
    • Secant ($\sec$): reciprocal of cosine.
    • Cotangent ($\cot$): reciprocal of tangent.
  • Important ratios are based on a triangle:
    • sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
    • cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
    • tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
Relationships of Trigonometric Functions
  • If you know two trigonometric ratios, you can derive the others using:
    • Reciprocal identities (e.g., csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)})
    • Quotient identities (e.g., tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)})
Calculator Usage
  • Ensure calculator is set to either degree or radian mode.
  • Use inverse trigonometric functions to find angles from their sine, cosine, or tangent values:
    • Inverse sine of $\frac{3}{5}$ gives you the angle for which sine equals $ rac{3}{5}$.
Solving Problems with Triangles
  • The Pythagorean theorem states that in a right triangle:
    a2+b2=c2a^2 + b^2 = c^2
  • Use known side lengths to find missing side lengths or angles using trigonometric functions.
  • For example, in a triangle with sides $3$ and $4$, the hypotenuse $c$ can be found:
    32+42=c23^2 + 4^2 = c^2
    which gives $c = 5$.
Applying Trigonometric Ratios
  • When determining unknown angles or side lengths, set up trigonometric equations based on the relationships of the sides of the triangle and solve for the required value using algebra or trigonometric identities.
  • Use the framework of the sine, cosine, or tangent ratios to express unknown sides or angles in terms of known quantities for finding solutions efficiently.