Consider the sequence whose first five terms are shown. -59 ,-67 ,-65 ,-53 ,-31 ,... The first term is ____ To calculate the sec
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<span>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 </span>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
, that is, the line in green.
a.2 <span>Explain why its length is sec θ.
We know these relationships:
(1)
(2) </span>
<span>
(3) </span>
<span>
Triangles </span>ΔOFC and ΔOBD are similar, so it is true that:
<span>
</span>∴
<span>
b.1 </span>Next, find cot θ
The segment whose length is cot θ is shown in Figure 3, this length is the segment
, that is, the line in pink.
b.2 <span>Use the representation of tangent as a clue for what to show for cotangent.
</span>
It's true that:
But:
Then:
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:
c. find csc θ in your diagram.
The segment whose length is csc θ is shown in Figure 4, this length is the segment
, that is, the line in green.
The representation of the diagram is shown if Figure 1. There's a relationship between </span>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
a.2 <span>Explain why its length is sec θ.
We know these relationships:
(1)
(2) </span>
<span>
(3) </span>
<span>
Triangles </span>ΔOFC and ΔOBD are similar, so it is true that:
</span>∴
b.1 </span>Next, find cot θ
The segment whose length is cot θ is shown in Figure 3, this length is the segment
b.2 <span>Use the representation of tangent as a clue for what to show for cotangent.
</span>
It's true that:
But:
Then:
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:
c. find csc θ in your diagram.
The segment whose length is csc θ is shown in Figure 4, this length is the segment