Educator

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.
• 

tanθ = [5/6], tanϕ = [6/5], tan[(π)/2] = undefined, cotθ = [6/5], cotϕ = [5/6], cot[(π)/2] = 0

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.
• 

tanθ = [9/11], tanϕ = [11/9], tan[(π)/2] = undefined, cotθ = [11/9], cotϕ = [9/11], cot[(π)/2] = 0

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.
• 

tanθ = [(√{95} )/7], tanϕ = [(7√{95} )/95], tan[(π)/2] = undefined, cotϕ = [(7√{95} )/95], cotϕ = [(√{95} )/7], cot[(π)/2] = 0

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.
• 

tanθ = [(5√{39} )/39], tanϕ = [(√{39} )/5], tan[(π)/2] = undefined, cotθ = [(√{39} )/5], cotϕ = [(5√{39} )/39], cot[(π)/2] = 0

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.
• 

tanθ = [(√5 )/2], tanϕ = [(2√5 )/5], tan[(π)/2] = undefined, cotθ = [(2√5 )/5], cotϕ = [(√5 )/2], cot[(π)/2] = 0

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

Angle	sin	cos	tan	cot
[(3π)/4]	[(√2 )/2]	[(√2 )/2]	− 1	− 1
[(4π)/3]	− [(√3 )/2]	− [1/2]	√3	[(√3 )/3]
[(11π)/6]	− [1/2]	[(√3 )/2]	− [(√3 )/3]	− √3

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]

1. Intro
2. Amplitude and Period of a Sine Wave
• Sine Wave Graph
• Amplitude: Distance from Middle to Peak
• Peak: Distance from Peak to Peak
3. Phase Shift and Vertical Shift
• Phase Shift: Distance Shifted Horizontally
• Vertical Shift: Distance Shifted Vertically
4. Example 1: Amplitude/Period/Phase and Vertical Shift
5. Example 2: Amplitude/Period/Phase and Vertical Shift
6. Example 3: Find Sine Wave Given Attributes
7. Extra Example 1: Amplitude/Period/Phase and Vertical Shift
8. Extra Example 2: Find Cosine Wave Given Attributes