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### Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

Main definitions and formulas:

• The amplitude of a sine wave is the vertical distance from the middle of the waves to the peaks (or from the middle to the valleys). In the equations above, it is given by | A| .
• The period of a sine wave is the horizontal distance for the wave to do one complete cycle from one peak to the next peak. In the equations above, it is given by (2π/B).
• The phase shift of a sine wave is the horizontal distance the wave is shifted from the traditional starting position. In the equations above, it is given by − (C/B).
• The vertical shift of a sine wave is the vertical distance that the middle of the wave is shifted from the x-axis. In the equations above, it is given by D.

Example 1:

Identify the amplitude, period, phase shift, and vertical shift of the following function, and graph the function:
 3 cos(4x+π ) + 2

Example 2:

Identify the amplitude, period, phase shift, and vertical shift of the following function, and graph the function:
 − 2 sin ( 2x− π 3 )

Example 3:

Find a sine wave with amplitude 2, period 4π , phase shift (π/2), and vertical shift 1. Graph the function.

Example 4:

Identify the amplitude, period, phase shift, and vertical shift of the following function, and graph the function:
 4 sin ( 2x+ π 2 ) − 1

Example 5:

Find a cosine wave with amplitude 2, period 3π , phase shift (π/2), and vertical shift -2. Graph the function.

## A triangle has short sides of length 5 and 6. Find the tangents and cotangents of all the angles in the triangle.

• Recall: SOHCAHTOA, tanθ = [Opposite/Adjacent], cotθ = [Adjacent/Opposite]
• First calculate the missing leg of the triangle by using the Pythagorean Theorem: a2 + b2 = c2
• 52 + 62 = x2 ⇒ 25 + 36 = x2 ⇒ 61 = x2 ⇒ √{61} = x
• Draw a triangle in which the sides and angles are labeled so tangent and cotangent can be determined.

## A triangle has short sides of length 9 and 11. Find the tangents and cotangents of all the angles in the triangle.

• Recall: SOHCAHTOA, tanθ = [Opposite/Adjacent], cotθ = [Adjacent/Opposite]
• First calculate the missing leg of the triangle by using the Pythagorean Theorem: a2 + b2 = c2
• 92 + 112 = x2 ⇒ 81 + 121 = x2 ⇒ 202 = x2 ⇒ √{202} = x
• Draw a triangle in which the sides and angles are labeled so tangent and cotangent can be determined.

## A triangle has hypotenuse of length 12 and side of length 7. Find the tangents and cotangents of all the angles in the triangle.

• Recall: SOHCAHTOA, tanθ = [Opposite/Adjacent], cotθ = [Adjacent/Opposite]
• First calculate the missing leg of the triangle by using the Pythagorean Theorem: a2 + b2 = c2
• x2 + 72 = 122 ⇒ x2 + 49 = 144 ⇒ 95 = x2 ⇒ √{95} = x
• Draw a triangle in which the sides and angles are labeled so tangent and cotangent can be determined.

## A triangle has hypotenuse of length 8 and side of length 5. Find the tangents and cotangents of all the angles in the triangle.

• Recall: SOHCAHTOA, tanθ = [Opposite/Adjacent], cotθ = [Adjacent/Opposite]
• First calculate the missing leg of the triangle by using the Pythagorean Theorem: a2 + b2 = c2
• x2 + 52 = 82 ⇒ x2 + 25 = 64 ⇒ 39 = x2 ⇒ √{39} = x
• Draw a triangle in which the sides and angles are labeled so tangent and cotangent can be determined.

## A triangle has hypotenuse of length 6 and side of length 4. Find the tangents and cotangents of all the angles in the triangle.

• Recall: SOHCAHTOA, tanθ = [Opposite/Adjacent], cotθ = [Adjacent/Opposite]
• First calculate the missing leg of the triangle by using the Pythagorean Theorem: a2 + b2 = c2
• x2 + 42 = 62 ⇒ x2 + 16 = 36 ⇒ 20 = x2 ⇒ 2√5 = x
• Draw a triangle in which the sides and angles are labeled so tangent and cotangent can be determined.

## Find the tangent and cotangent of the following angles: [(3π)/4],[(4π)/3],[(11π)/6]

• First plot all three angles on the unit circle. Notice that [(3π)/4] uses the 45° − 45° − 90° triangle and [(4π)/3],[(11π)/6] both use the 30° − 60° − 90° triangle.
• Use the mnemonic ASTC to determine which ones are positive and which are negative

## Sketch the graph of f(x) = tan 2x. Label the zeroes and asymptotes.

• First find two consecutive asymptotes by solving the following equations: 2x = − [(π)/2] and 2x = [(π)/2]
• So, x = − [(π)/4] and x = [(π)/4] are two asymptotes
• Use the asymptotes and sketch the tangent graph
• Zeroes are − [(π)/2], 0, [(π)/2], π, [(3π)/2], 2π, Asymptotes are [( − π)/4], [(π)/4], [(3π)/4], [(5π)/4], [(7π)/4], [(9π)/4], ...

## Sketch the graph of f(x) = tan [x/2]. Label the zeroes and asymptotes.

• First find two consecutive asymptotes by solving the following equations: [x/2] = − [(π)/2] and [x/2] = [(π)/2]
• So, x = - π and x = π are two asymptotes
• Use the asymptotes and sketch the tangent graph
• Zeroes are 0, 2π, Asymptotes are - π, π, 3π

## Sketch the graph of f(x) = cot [x/3]. Label the zeroes and asymptotes.

• First find two consecutive asymptotes by solving the following equations: [x/3] = 0 and [x/3] = π
• So, x = 0 and x = 3π are two asymptotes
• Use the asymptotes and sketch the tangent graph
• Zeroes are − [(3π)/2], [(3π)/2], [(9π)/2], Asymptotes are 0, π, 3π

## Sketch the graph of f(x) = cot 4x. Label the zeroes and asymptotes.

• First find two consecutive asymptotes by solving the following equations: 4x = 0 and 4x = π
• So, x = 0 and x = [(π)/4] are two asymptotes
• Use the asymptotes and sketch the tangent graph
• Zeroes are − [(π)/8], [(π)/8], [(π)/2], Asymptotes are − [(π)/4], 0, [(π)/4], [(3π)/4]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.