in exercises 21 - 32, evaluate the function without using technology. example 4 21. sin 45° 22. cot 60° 23…

in exercises 21 - 32, evaluate the function without using technology. example 4 21. sin 45° 22. cot 60° 23. sec 135° 24. tan 240° 25. sin(-150°) 26. csc(-420°) 27. cos(π/4) 28. tan(π/6) 29. tan(-3π/4) 30. cot(-8π/3) 31. cos(7π/4) 32. sec(11π/6)

in exercises 21 - 32, evaluate the function without using technology. example 4 21. sin 45° 22. cot 60° 23. sec 135° 24. tan 240° 25. sin(-150°) 26. csc(-420°) 27. cos(π/4) 28. tan(π/6) 29. tan(-3π/4) 30. cot(-8π/3) 31. cos(7π/4) 32. sec(11π/6)

Answer

Answer:

  1. $\sec135^{\circ}=-\sqrt{2}$

Explanation:

Step1: Recall secant - cosine relation

$\sec\theta=\frac{1}{\cos\theta}$, so $\sec135^{\circ}=\frac{1}{\cos135^{\circ}}$

Step2: Determine the reference angle

The reference angle for $135^{\circ}$ is $180 - 135=45^{\circ}$. And $135^{\circ}$ is in the second - quadrant where cosine is negative.

Step3: Find the cosine value

We know that $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, so $\cos135^{\circ}=-\frac{\sqrt{2}}{2}$

Step4: Calculate the secant value

$\sec135^{\circ}=\frac{1}{\cos135^{\circ}}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$