Lesson 6

The Pythagorean Identity (Part 2)

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Problem 1

The picture shows angles \(A\) and \(B\). Explain why \(\sin(B) = \text- \sin(A)\) and why \(\cos(B) = \text-\cos(A)\).

Circle on a coordinate plane, center at the origin, radius 1. A diameter through the first and third quadrants. Acute angle formed with the positive x axis has measure A. Larger angle has measure B.

Solution

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Problem 2

Which statements are true? Select all that apply.

A:

\(\sin(\theta) > 0\) for an angle \(\theta\) in quadrant 2

B:

\(\cos(\theta) > 0\) for an angle \(\theta\) in quadrant 2

C:

\(\tan(\theta) > 0\) for an angle \(\theta\) in quadrant 2

D:

\(\sin(\theta) > 0\) for an angle \(\theta\) in quadrant 3

E:

\(\cos(\theta) > 0\) for an angle \(\theta\) in quadrant 3

F:

\(\tan(\theta) > 0\) for an angle \(\theta\) in quadrant 3

Solution

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Problem 3

The tangent of an angle satisfies \(\tan(\theta) = 10\).

  1. Which quadrant could \(\theta\) lie in? Explain how you know.
  2. Estimate the possible value(s) of \(\theta\). Explain your reasoning.

Solution

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Problem 4

Evaluate each of the following:

  1. \(\tan\left(\frac{5\pi}{4}\right)\)
  2. \(\sin\left(\frac{3\pi}{2}\right)\)
  3. \(\cos\left(\frac{7\pi}{4}\right)\)

Solution

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Problem 5

The sine of an angle \(\theta\) in the second quadrant is \(0.6\). What is \(\tan(\theta)\)? Explain how you know. 

Solution

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Problem 6

Triangle \(ABC\) is an isosceles right triangle in the unit circle.

A circle with center A at the origin of an x y plane.
  1. Explain why \(\sin(A) = \cos(A)\).
  2. Use the Pythagorean Theorem to explain why \(2(\sin(A))^2 = 1\).

Solution

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(From Unit 6, Lesson 5.)

Problem 7

Triangle \(DEF\) is similar to triangle \(ABC\). The scale factor going from \(\triangle DEF\) to \(\triangle ABC\) is 3.

Two triangles. First, A, B C with B is a right angle. Second, D E F, E is a right angle.
  1. Explain why the length of segment \(AB\) is 3 times the length of segment \(DE\) and the length of segment \(BC\) is 3 times the length of segment \(EF\).
  2. Explain why \(\sin(A) = \sin(D)\).

Solution

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(From Unit 6, Lesson 2.)

Problem 8

Which of the following is true for angle \(\theta\)? Select all that apply.

A circle with radius 1 inscribed on a coordinate plane, centered at the origin. Point P is on the circle in the third quadrant. Sector is labeled theta.
A:

\(\sin(\theta) < 0\)

B:

\(\sin(\theta) > 0\)

C:

\(\cos(\theta) < 0\)

D:

\(\cos(\theta) > 0\)

E:

\(\sin(\theta) > \cos(\theta)\)

F:

\(\sin(\theta) < \cos(\theta)\)

Solution

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(From Unit 6, Lesson 5.)