Find sec θ and csc θ. If tan θ =3 and cos θ>0. Please helpp, i don't get identities

cosØ>0 tells me Ø is in either I or IV

tanØ = 3 tells me Ø is in I or III
So Ø must be in the 1st quadrant

tanØ = 3 = 3/1 = opposite/adjacent
so construct your triangle with those sides,
easy to see that the hypotenuse is √10

so cosØ = 1/√10 ----> secØ = √10
and sinØ = 3/√10 ---> cscØ = √10/3

Well, well, well, if tangent is a mystery to you, let me shed some light on this conundrum. Tangent is defined as the ratio of the sine of an angle to the cosine of the same angle. So, if tan θ = 3 and we know that cos θ is greater than 0, we can use the Pythagorean identity to find the remaining trigonometric functions.

First, let's find the sine of θ. We know that tan θ = sin θ / cos θ, so we can rewrite this as sin θ = tan θ * cos θ. Plugging in the values we have, sin θ = 3 * cos θ.

Now, we can use the Pythagorean identity, sin^2 θ + cos^2 θ = 1, to find the value of sin^2 θ. Since we know that cos θ is greater than 0, sin^2 θ = 1 - cos^2 θ. Plugging in the value we found earlier, sin^2 θ = 1 - (3 * cos θ)^2.

Now, we can solve for sin θ using the positive square root since sine is positive when tangent is positive and cosine is positive. Finally, we can find sec θ and csc θ using the reciprocal identities.

I hope that wasn't too much math juggling for you! Keep those questions coming!

To find sec θ and csc θ, we can use the following trigonometric identities:

sec θ = 1/cos θ
csc θ = 1/sin θ

We are given that tan θ = 3 and cos θ > 0.

Since tan θ = sin θ / cos θ, and we are given that tan θ = 3, we can write:

3 = sin θ / cos θ

Rearranging the equation, we can multiply both sides by cos θ:

3 * cos θ = sin θ

Now, we can use the Pythagorean identity sin²θ + cos²θ = 1 to find sin θ. Since cos θ > 0, we can choose the positive square root:

cos²θ = 1 - sin²θ
cos θ = √(1 - sin²θ)
cos θ = √(1 - 3²) = √(1 - 9) = √(-8)

Since cos θ is positive, we cannot take a square root of a negative number, which means there is no solution for θ that satisfies the given conditions.

Therefore, sec θ and csc θ cannot be determined in this case.

Sure! To find sec θ and csc θ, we can use the following trigonometric identities:

1. sec θ = 1 / cos θ
2. csc θ = 1 / sin θ

We are given that tan θ = 3 and cos θ > 0.

First, let's use the given information to find sin θ.

We know that tan θ = sin θ / cos θ. Plugging in the given value of tan θ as 3, we get:

3 = sin θ / cos θ

Since we want cos θ to have a positive value, we can set it to be 1. So, cos θ = 1.

Now, we can solve for sin θ:

3 = sin θ / 1
sin θ = 3

Using the Pythagorean identity, we can find the length of the hypotenuse of the right triangle formed by sin θ and cos θ:

sin^2 θ + cos^2 θ = 1

(3)^2 + (1)^2 = 1
9 + 1 = 10

Taking the square root of both sides, we get:

sin θ = √10

Now, we can find sec θ and csc θ.

sec θ = 1 / cos θ
Since cos θ is 1, sec θ = 1 / 1 = 1.

csc θ = 1 / sin θ
Since sin θ is √10, csc θ = 1 / √10.

Thus, we have found that sec θ = 1 and csc θ = 1 / √10 (or you can also write it as √10 / 10).

I hope this helps! Let me know if you have any further questions.