Given sec θ =(square root) 10 and tan θ = 3, determine the following, csc(90° − θ)

Well, well, well, isn't trigonometry just full of surprises? Let's see what we can do here.

We know that sec(theta) is the reciprocal of cosine(theta). Since sec(theta) is equal to the square root of 10, cosine(theta) must be equal to its reciprocal, which is 1 over the square root of 10. And we all know that the square root of 10, well, it's just not a 'rational' guy, you know?

Now, once we know the value of cosine(theta), we can use the Pythagorean identity to find sine(theta). Remember the good old Pythagorean theorem? It's all a^2 + b^2 = c^2. In this case, sine squared(theta) plus cosine squared(theta) equals one. So, with a little substitution, we can find sine(theta).

Now, csc(90° - theta) is just the reciprocal of sine(90° - theta). And since we know the value of sine(theta), we can use some trigonometric identity tricks and find the value of csc(90° - theta). But hey, let's not make this too complicated, let's have some fun with it!

If you remember that sine(90° - theta) is equal to cosine(theta), then we can replace it and just say that csc(90° - theta) is equal to the reciprocal of cosine(theta). We already found out that cosine(theta) is 1 over the square root of 10, so, drum roll, please, the value of csc(90° - theta) is just the reciprocal of that. Simplify, and you'll have your answer!

Reciprocal of 1 over the square root of 10? Well, that's simple! Just multiply the numerator and denominator by the square root of 10. So, csc(90° - theta) is equal to the square root of 10 over 1, which is just, you guessed it, the square root of 10!

Voila! csc(90° - theta) = √10. And they say math isn't fun!

To find csc(90° − θ), we need to find the value of sin(90° − θ) and then take its reciprocal.

First, let's find sin(90° − θ):
We know that sin(90° − θ) = cos(θ).

Given that sec θ = √10, we can use the identity sec^2 θ - 1 = tan^2 θ, to find the value of cos θ:

sec^2 θ - 1 = tan^2 θ
(√10)^2 - 1 = 3^2
10 - 1 = 9
9 = 9

So, cos θ = √9 = 3.

Since sin(90° − θ) = cos(θ) = 3, we can now find csc(90° − θ) as the reciprocal of sin(90° − θ):

csc(90° − θ) = 1/sin(90° − θ) = 1/3.

Therefore, csc(90° − θ) = 1/3.

To determine csc(90° - θ), we need to find the value of sec θ first.

Given that sec θ = √10, we can use the reciprocal identity to find the value of cos θ. The reciprocal identity states that sec θ = 1/cos θ.

So, we have:
√10 = 1/cos θ

To find cos θ, we can take the reciprocal of √10:
cos θ = 1/√10

We can rationalize the denominator by multiplying the numerator and denominator by √10:
cos θ = √10/(√10 * √10)
cos θ = √10/10
cos θ = √10/10

Since tan θ = 3, we can use the identity tan θ = sin θ / cos θ to find the value of sin θ.

So, we have:
3 = sin θ / (√10/10)
Multiplying both sides by (√10/10):
3 * (√10/10) = sin θ
(3√10)/10 = sin θ

Now, we can use the identity csc θ = 1/sin θ to find the value of csc(90° - θ).

csc(90° - θ) = 1/sin(90° - θ)

We know that sin(90° - θ) = sin 90° * cos θ - cos 90° * sin θ

The value of sin 90° is 1 and the value of cos 90° is 0. So, we have:
csc(90° - θ) = 1/(1 * √10/10) - 0 * (√10/10)
csc(90° - θ) = 1/(√10/10)
csc(90° - θ) = 10/√10

To rationalize the denominator, we can multiply both the numerator and denominator by √10:
csc(90° - θ) = (10/√10) * (√10/√10)
csc(90° - θ) = 10√10/10
csc(90° - θ) = √10

Therefore, csc(90° - θ) = √10.

The co- in cosine, cotangent, cosecant means "of the complementary angle."

csc(90-θ) = secθ

always