Question

find the \\(\sec(t)\\), \\(\csc(t)\\), \\(\cot(t)\\) if \\(t\\) is the angle in blue.
draw
show your work here
\\(\sec(t) =\\)
\\(\csc(t) =\\)
\\(\cot(t) =\\)

Explanation:

Step1: Recall Trigonometric Definitions

For a right triangle, we know the following definitions:

• \(\sec(t)=\frac{\text{Hypotenuse}}{\text{Adjacent}}\)
• \(\csc(t)=\frac{\text{Hypotenuse}}{\text{Opposite}}\)
• \(\cot(t)=\frac{\text{Adjacent}}{\text{Opposite}}\)

From the right triangle, for angle \(t\):

• Hypotenuse \(= 15.62\)
• Adjacent side (to angle \(t\)) \(= 12\)
• Opposite side (to angle \(t\)) \(= 10\)

Step2: Calculate \(\sec(t)\)

Using the formula for \(\sec(t)\):
\(\sec(t)=\frac{\text{Hypotenuse}}{\text{Adjacent}}=\frac{15.62}{12}\approx1.3017\) (or we can check if \(12 - 10 - 15.62\) is a Pythagorean triple: \(12^{2}+ 10^{2}=144 + 100 = 244\), and \(15.62^{2}\approx244\) (since \(15.62\times15.62=(15 + 0.62)^{2}=15^{2}+2\times15\times0.62+0.62^{2}=225+18.6 + 0.3844 = 243.9844\approx244\)), so it is a right triangle. So \(\sec(t)=\frac{15.62}{12}=\frac{1562}{1200}=\frac{781}{600}\approx1.3017\) (or we can use exact fraction, but since \(15.62=\frac{1562}{100}=\frac{781}{50}\), so \(\frac{781/50}{12}=\frac{781}{600}\approx1.3017\))

Step3: Calculate \(\csc(t)\)

Using the formula for \(\csc(t)\):
\(\csc(t)=\frac{\text{Hypotenuse}}{\text{Opposite}}=\frac{15.62}{10}=1.562\) (or \(\frac{781}{50}\div10=\frac{781}{500}=1.562\))

Step4: Calculate \(\cot(t)\)

Using the formula for \(\cot(t)\):
\(\cot(t)=\frac{\text{Adjacent}}{\text{Opposite}}=\frac{12}{10}=1.2\)

Answer:

\(\sec(t)=\frac{781}{600}\approx1.30\) (or \(\frac{15.62}{12}\approx1.30\))
\(\csc(t)=1.562\) (or \(\frac{15.62}{10}\))
\(\cot(t)=1.2\) (or \(\frac{12}{10}\))

(If we want exact values, since \(15.62 = \sqrt{12^{2}+10^{2}}=\sqrt{244}\), so \(\sec(t)=\frac{\sqrt{244}}{12}=\frac{2\sqrt{61}}{12}=\frac{\sqrt{61}}{6}\approx\frac{7.81}{6}\approx1.30\), \(\csc(t)=\frac{\sqrt{244}}{10}=\frac{2\sqrt{61}}{10}=\frac{\sqrt{61}}{5}\approx1.56\), \(\cot(t)=\frac{12}{10}=\frac{6}{5}=1.2\))

So:
\(\sec(t)=\frac{\sqrt{61}}{6}\approx1.30\) (or \(1.30\) if decimal is preferred)
\(\csc(t)=\frac{\sqrt{61}}{5}\approx1.56\) (or \(1.56\))
\(\cot(t)=\frac{6}{5}=1.2\)

Final answers (using the given numbers directly):
\(\sec(t)=\frac{15.62}{12}\approx1.30\) (or \(\frac{781}{600}\))
\(\csc(t)=\frac{15.62}{10}=1.562\)
\(\cot(t)=\frac{12}{10}=1.2\)

So filling in the boxes:
\(\sec(t)=\boldsymbol{\frac{15.62}{12}\approx1.30}\) (or \(\boldsymbol{1.30}\) or \(\boldsymbol{\frac{781}{600}}\))
\(\csc(t)=\boldsymbol{1.562}\)
\(\cot(t)=\boldsymbol{1.2}\)