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The diagram shows how cos θ, sin θ, and tan θ relate to the unit circle. Copy the diagram and show how sec θ, csc θ, and cot θ r

DIA [1.3K]

Copy the diagram and show how sec θ, csc θ, and cot θ relate to the unit circle.

The representation of the diagram is shown if Figure 1. There's a relationship between sec θ, csc θ, and cot θ related the unit circle. Lines green, blue and pink show the relationship.

a.1 First, find in the diagram a segment whose length is sec θ.

The segment whose length is sec θ is shown in Figure 2, this length is the segment $\overline{OF}$ , that is, the line in green.

a.2 Explain why its length is sec θ.

We know these relationships:

(1) $sin \theta=\frac{\overline{BD}}{\overline{OB}}=\frac{\overline{BD}}{r}=\frac{\overline{BD}}{1}=\overline{BD}$

(2) $cos \theta=\frac{\overline{OD}}{\overline{OB}}=\frac{\overline{OD}}{r}=\frac{\overline{OD}}{1}=\overline{OD}$

(3) $tan \theta=\frac{\overline{FD}}{\overline{OC}}=\frac{\overline{FC}}{r}=\frac{\overline{FC}}{1}=\overline{FC}$

Triangles ΔOFC and ΔOBD are similar, so it is true that:

$\frac{\overline{FC}}{\overline{OF}}= \frac{\overline{BD}}{\overline{OB}}$ 

∴ $\overline{OF}= \frac{\overline{FC}}{\overline{BD}}= \frac{tan \theta}{sin \theta}= \frac{1}{cos \theta} \rightarrow \boxed{sec \theta= \frac{1}{cos \theta}}$ 

b.1 Next, find cot θ

The segment whose length is cot θ is shown in Figure 3, this length is the segment $\overline{AR}$ , that is, the line in pink.

b.2 Use the representation of tangent as a clue for what to show for cotangent.

It's true that:

$\frac{\overline{OS}}{\overline{OC}}= \frac{\overline{SR}}{\overline{FC}}$

But:

$\overline{SR}=\overline{OA}$
$\overline{OS}=\overline{AR}$

Then:

$\overline{AR}= \frac{1}{\overline{FC}}= \frac{1}{tan\theta} \rightarrow \boxed{cot \theta= \frac{1}{tan \theta}}$

b.3 Justify your claim for cot θ.

As shown in Figure 3, θ is an internal angle and ∠A = 90°, therefore ΔOAR is a right angle, so it is true that:

$cot \theta= \frac{\overline{AR}}{\overline{OA}}=\frac{\overline{AR}}{r}=\frac{\overline{AR}}{1} \rightarrow \boxed{cot \theta=\overline{AR}}$

c. find csc θ in your diagram.

The segment whose length is csc θ is shown in Figure 4, this length is the segment $\overline{OR}$ , that is, the line in green.

3 0

4 years ago

The more surface area an ice cube has, the more quickly it melts to cool a beverage. mark states that the ideal ice cube would b

Ber [7]

It is true that the more surface area an ice cube has, the more quickly it melts to cool a beverage. By making the length of the prism shorter by 1 and the height longer by 2, the resulting prism will look like a 2D object floating on the beverage. Thus, it will cool it faster.

The correct answer is: 1 by 2 by 4 or 1 x 2 x 4

5 0

4 years ago

Read 2 more answers

What is the solution to the equation below? logg x = 4

Darya [45]

Anwser is b

Because I just did the same quiz

5 0

3 years ago

Help and please explain!

UkoKoshka [18]

Answer:

a. 146

Step-by-step explanation:

{4*[72-6*(3+2)]}-22

=[4*(72-6*5)]-22

=[4*(72-30)]-22

=(4*42)-22

=168-22

=146

4 0

3 years ago

Which is the side length of a cube with a volume of 64 m3?

Karo-lina-s [1.5K]

Which is the side length of a cube with a volume of 64 m3?

Volume: 64m³

a^3 = 64
\sqrt[3]{64} [/tex] = 4 m
↑
the answer
Check
4 * 4 * 4 = 64 m³

4 0

4 years ago