find the \\(\\sec(t)\\), \\(\\csc(t)\\), \\(\\cot(t)\\) if \\(t\\) is the angle in blue.\ndraw\nshow your…

Explanation:

Step1: Recall trigonometric identities

We know that for a right triangle, (\sec(t)=\frac{1}{\cos(t)}), (\csc(t)=\frac{1}{\sin(t)}), (\cot(t)=\frac{1}{\tan(t)}). Also, (\cos(t)=\frac{\text{adjacent}}{\text{hypotenuse}}), (\sin(t)=\frac{\text{opposite}}{\text{hypotenuse}}), (\tan(t)=\frac{\text{opposite}}{\text{adjacent}}).

For angle (t):

• Adjacent side (to (t)): (10)
• Opposite side (to (t)): (12)
• Hypotenuse: (15.62) (given)

Step2: Calculate (\sec(t))

(\cos(t)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{10}{15.62}) So, (\sec(t)=\frac{1}{\cos(t)}=\frac{15.62}{10} = 1.562)

Step3: Calculate (\csc(t))

(\sin(t)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{12}{15.62}) So, (\csc(t)=\frac{1}{\sin(t)}=\frac{15.62}{12}\approx1.3017) (or keep as (\frac{15.62}{12}), but let's compute the decimal: (15.62\div12\approx1.3017))

Step4: Calculate (\cot(t))

(\tan(t)=\frac{\text{opposite}}{\text{adjacent}}=\frac{12}{10} = 1.2) So, (\cot(t)=\frac{1}{\tan(t)}=\frac{10}{12}=\frac{5}{6}\approx0.8333)

Answers:

(\sec(t)=\boldsymbol{1.562})

(\csc(t)=\boldsymbol{\frac{15.62}{12}\approx1.302}) (or (\frac{781}{600}) if we use exact fraction, but decimal is fine here)

(\cot(t)=\boldsymbol{\frac{5}{6}\approx0.833})

(Note: If we check the Pythagorean theorem: (10^2 + 12^2=100 + 144 = 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 the triangle is valid.)