Question

determining trigonometric function values on the...
when try to determine the function values of cosecant, secant,
and cotangent we do need to be a little more careful.
remembering the relationships among the trigonometric
functions is important.

1. ( sec (102.5^{circ})approx - 4.6202 )

recall
( sec\theta=\frac{1}{cos\theta} )

1. ( csc (432^{circ}) )

recall
( csc\theta=\frac{1}{sin\theta} )

1. ( cot(-23.45^{circ}) )

recall
( cot\theta=\frac{1}{\tan\theta}=\frac{cos\theta}{sin\theta} )
find the exact value.
( \tan(750^{circ})=)
( cot(750^{circ})=)

Explanation:

Step1: Find the coterminal angle

Since the trigonometric functions have a period of \(360^{\circ}\), we can find a coterminal angle for \(750^{\circ}\) by subtracting multiples of \(360^{\circ}\).
\(750^{\circ}-2\times360^{\circ}=750^{\circ} - 720^{\circ}=30^{\circ}\)

Step2: Calculate \(\tan(750^{\circ})\)

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for \(\theta = 30^{\circ}\), \(\sin(30^{\circ})=\frac{1}{2}\), \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\)
\(\tan(750^{\circ})=\tan(30^{\circ})=\frac{\sin(30^{\circ})}{\cos(30^{\circ})}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)

Step3: Calculate \(\cot(750^{\circ})\)

We know that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), and for \(\theta = 30^{\circ}\), \(\sin(30^{\circ})=\frac{1}{2}\), \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\)
\(\cot(750^{\circ})=\cot(30^{\circ})=\frac{\cos(30^{\circ})}{\sin(30^{\circ})}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}\)

Answer:

\(\tan(750^{\circ})=\frac{\sqrt{3}}{3}\)
\(\cot(750^{\circ})=\sqrt{3}\)