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pishuonlain [190]
3 years ago
13

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not u

se a calculator. -135^{\circ}−135
Mathematics
1 answer:
Solnce55 [7]3 years ago
4 0

Answer:

Step-by-step explanation:

Let's take a look at the given angle 135°

The sketch of the angle which corresponds to -\dfrac{3\pi}{4} unit circle and can be seen in the attached image below;

The trigonometric ratios are as follows for an angle θ on the unit circle:

Trigonometric ratio           related ratio on coordinate axes

sin θ                                     \dfrac{y}{1}

cos θ                                    \dfrac{x}{1}

tan θ                                     \dfrac{y}{x}

csc θ                                     \dfrac{1}{y}

sec θ                                     \dfrac{1}{x}

cot θ                                      \dfrac{x}{y}

From the sketch of the image attached below;

The six trigonometric ratio for 135° can be expressed as follows:

sin (-\dfrac{3\pi}{4})= \dfrac{y}{1}

sin (-\dfrac{3\pi}{4})=- \dfrac{\sqrt{2}}{2}

cos (-\dfrac{3\pi}{4})= \dfrac{x}{1}

cos (-\dfrac{3\pi}{4})= -\dfrac{\sqrt{2}}{2}

tan (-\dfrac{3\pi}{4})= \dfrac{y}{x}

tan (-\dfrac{3\pi}{4})= \dfrac{-\dfrac{\sqrt{2}}{2}}{-\dfrac{\sqrt{2}}{2}}

tan (-\dfrac{3\pi}{4})= -\dfrac{\sqrt{2}}{2}} \times {-\dfrac{2}{\sqrt{2}}

tan (-\dfrac{3\pi}{4})= 1

csc (-\dfrac{3\pi}{4})= \dfrac{1}{y} \\ \\ csc (-\dfrac{3\pi}{4})=\dfrac{1}{-\dfrac{\sqrt{2}}{2}} \\ \\ csc=1 \times -\dfrac{2}{\sqrt{2}} \\ \\csc =-\sqrt{2}

sec (-\dfrac{3 \pi}{4})=\dfrac{1}{x} \\ \\  sec = \dfrac{1}{(-\dfrac{\sqrt{2}}{2})} \\ \\ sec = 1 \times -\dfrac{2}{\sqrt{2}}  \\ \\  sec = - \sqrt{2}

cot(-\dfrac{3 \pi}{4}) = \dfrac{x}{y} \\ \\ cot(-\dfrac{3 \pi}{4}) = \dfrac{-\dfrac{\sqrt{2}}{2} }{-\dfrac{\sqrt{2}}{2}} \\ \\  cot(-\dfrac{3 \pi}{4})=  -\dfrac{\sqrt{2}}{2} } \times {-\dfrac{2}{\sqrt{2}}} \\ \\  cot (-\dfrac{3 \pi}{4}) = 1

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