How do you find all six trigonometric function of #theta# if the point (3,4) is on the terminal side of #theta#?

Answer 1
The terminal side, containing point (3, 4) is located in Quadrant 1. Call t the angle: #tan t = y/x = 4/3# #cos^2 t = 1/(1 + tan^2 t) = 1/(1 + 16/9) = 9/25# #cos t = 3/5# (t in Quadrant 1 --> cos t is positive) #sin^2 t = 1 - cos^2 t = 1 - 9/25 = 16/25# #sin t = 4/5# (t in Quadrant 1 --> sin t is positive) #tan t = 4/3# #cot t = 1/(tan t) = 3/4# #sec t = 1/(cos t) = 5/3# #csc t = 1/(sin t) = 5/4#
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Answer 2

To find all six trigonometric functions of ( \theta ) if the point (3, 4) is on the terminal side of ( \theta ), we can use the coordinates of the point to determine the values of sine, cosine, tangent, secant, cosecant, and cotangent.

Given that the point (3, 4) is on the terminal side of ( \theta ), we can form a right triangle with one side along the x-axis, another side along the y-axis, and the hypotenuse passing through the point (3, 4).

Using the Pythagorean theorem, we can find the length of the hypotenuse: [ r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 ]

Now, we can determine the trigonometric functions:

  1. Sine (sin): [ \sin(\theta) = \frac{y}{r} = \frac{4}{5} ]

  2. Cosine (cos): [ \cos(\theta) = \frac{x}{r} = \frac{3}{5} ]

  3. Tangent (tan): [ \tan(\theta) = \frac{y}{x} = \frac{4}{3} ]

  4. Secant (sec): [ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{3}{5}} = \frac{5}{3} ]

  5. Cosecant (csc): [ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{4}{5}} = \frac{5}{4} ]

  6. Cotangent (cot): [ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\frac{4}{3}} = \frac{3}{4} ]

Therefore, the values of the six trigonometric functions of ( \theta ) are: [ \sin(\theta) = \frac{4}{5}, \cos(\theta) = \frac{3}{5}, \tan(\theta) = \frac{4}{3}, \sec(\theta) = \frac{5}{3}, \csc(\theta) = \frac{5}{4}, \cot(\theta) = \frac{3}{4} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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