Question

intro to all six trig functions

getting started: remember soh-cah-toa
\\(\sin\theta = \frac{o}{h}\\)
\\(\cos\theta = \frac{a}{h}\\)
\\(\tan\theta = \frac{o}{a}\\)

\\(\csc\theta = \frac{1}{\sin\theta}\\)
\\(\sec\theta = \frac{1}{\cos\theta}\\)
\\(\cot\theta = \frac{1}{\tan\theta}\\)

all six trig functions from a right triangle:

you do #1
find the six trigonometric functions for the angle \\(\theta\\) in the right triangle with sides 3, 4, and 5.

you do #2
find the six trigonometric functions for the angle \\(\theta\\) in the right triangle with sides 20, 21, and 29.

Explanation:

Identify sides for You Do #1

For the first triangle, the angle \(\theta\) is at the top vertex.

• The side opposite to \(\theta\) is \(4\).
• The side adjacent to \(\theta\) is \(3\).
• The hypotenuse is \(5\).

Calculate ratios for You Do #1

Using the definitions of the six trigonometric functions:

• \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5}\)
• \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{3}{5}\)
• \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{4}{3}\)
• \(\csc\theta = \frac{1}{\sin\theta} = \frac{5}{4}\)
• \(\sec\theta = \frac{1}{\cos\theta} = \frac{5}{3}\)
• \(\cot\theta = \frac{1}{\tan\theta} = \frac{3}{4}\)

Identify sides for You Do #2

For the second triangle, the angle \(\theta\) is at the top vertex.

• The side opposite to \(\theta\) is \(21\).
• The side adjacent to \(\theta\) is \(20\).
• The hypotenuse is \(29\).

Calculate ratios for You Do #2

Using the definitions of the six trigonometric functions:

• \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{21}{29}\)
• \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{20}{29}\)
• \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{21}{20}\)
• \(\csc\theta = \frac{1}{\sin\theta} = \frac{29}{21}\)
• \(\sec\theta = \frac{1}{\cos\theta} = \frac{29}{20}\)
• \(\cot\theta = \frac{1}{\tan\theta} = \frac{20}{21}\)

Answer:

Question 1

For You Do #1:

• \(\sin\theta = \frac{4}{5}\)
• \(\cos\theta = \frac{3}{5}\)
• \(\tan\theta = \frac{4}{3}\)
• \(\csc\theta = \frac{5}{4}\)
• \(\sec\theta = \frac{5}{3}\)
• \(\cot\theta = \frac{3}{4}\)

Question 2

For You Do #2:

• \(\sin\theta = \frac{21}{29}\)
• \(\cos\theta = \frac{20}{29}\)
• \(\tan\theta = \frac{21}{20}\)
• \(\csc\theta = \frac{29}{21}\)
• \(\sec\theta = \frac{29}{20}\)
• \(\cot\theta = \frac{20}{21}\)