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Questions and Answers
Find csc x and cot x if cos x = -4/7 and sin x > 0.
Find csc x and cot x if cos x = -4/7 and sin x > 0.
csc x = (7 √(33))/33, cot x = (-4 √33)/33
When tan x = 4/3 where π ≤ x ≤ 3π/2, find cos 2x.
When tan x = 4/3 where π ≤ x ≤ 3π/2, find cos 2x.
-7/25
When cos x = -8/17 where π ≤ x ≤ 3π/2, find cos 2x.
When cos x = -8/17 where π ≤ x ≤ 3π/2, find cos 2x.
-161/289
Solve this equation for 0 ≤ x ≤ 2π: (√2)sin(x)csc(x) + 2csc(x) = 2sin(x) + 2csc(x).
Solve this equation for 0 ≤ x ≤ 2π: (√2)sin(x)csc(x) + 2csc(x) = 2sin(x) + 2csc(x).
Solve: cos^2(x) + 2 = 2cos(x) + 1.
Solve: cos^2(x) + 2 = 2cos(x) + 1.
What are the solutions to the equation 0 = 2 tan(x) - sec^2(x)?
What are the solutions to the equation 0 = 2 tan(x) - sec^2(x)?
Solve -csc^2(x) + 2 = 0.
Solve -csc^2(x) + 2 = 0.
If cos(x) = cos 2(x), what are the solutions?
If cos(x) = cos 2(x), what are the solutions?
Solve for x in the equation 5 + sin x = (10 + √3)/2.
Solve for x in the equation 5 + sin x = (10 + √3)/2.
Find x such that arctan(tan(3Ï€/4)) = x.
Find x such that arctan(tan(3Ï€/4)) = x.
What is x if arctan(sec(Ï€)) = x?
What is x if arctan(sec(Ï€)) = x?
Find x if arccos(sin(0)) = x.
Find x if arccos(sin(0)) = x.
What is x if arcsin(csc(Ï€/2)) = x?
What is x if arcsin(csc(Ï€/2)) = x?
Solve the triangle with c = 17 cm, a = 15 cm, b = 28 cm.
Solve the triangle with c = 17 cm, a = 15 cm, b = 28 cm.
Solve the triangle where angle C = 65 degrees, b = 35 yd, c = 32 yd.
Solve the triangle where angle C = 65 degrees, b = 35 yd, c = 32 yd.
Solve the triangle with angle C = 62 degrees, b = 15 mi, c = 10 mi.
Solve the triangle with angle C = 62 degrees, b = 15 mi, c = 10 mi.
What is cos(-7Ï€/12)?
What is cos(-7Ï€/12)?
Find tan(5Ï€/12).
Find tan(5Ï€/12).
Given u = and v = , find -u - v.
Given u = and v = , find -u - v.
Given u = and g = , find u + g.
Given u = and g = , find u + g.
Find the angle between the two vectors u = -6i - 3j and v = 2i - 8j.
Find the angle between the two vectors u = -6i - 3j and v = 2i - 8j.
Find the angle between the vectors u = and v = .
Find the angle between the vectors u = and v = .
Write the vector CD in component form where C = and D = .
Write the vector CD in component form where C = and D = .
Find the component form of the resultant vector if a = and find 2a.
Find the component form of the resultant vector if a = and find 2a.
What is the direction angle for AB where A = and B = ?
What is the direction angle for AB where A = and B = ?
Find the dot product of the vectors u = and v = .
Find the dot product of the vectors u = and v = .
Find the dot product of u = -8i - 5j and v = 7i - 4j.
Find the dot product of u = -8i - 5j and v = 7i - 4j.
State if the vector u = and v = is parallel, orthogonal, or neither.
State if the vector u = and v = is parallel, orthogonal, or neither.
State if the vector u = and v = is parallel, orthogonal, or neither.
State if the vector u = and v = is parallel, orthogonal, or neither.
Write the vector -√3 + i in trig form.
Write the vector -√3 + i in trig form.
Write the vector √6 - 3i√2 in trig form.
Write the vector √6 - 3i√2 in trig form.
Write in rectangular form: 4(cos 60 + i sin 60).
Write in rectangular form: 4(cos 60 + i sin 60).
Write in rectangular form: 4(cos(4Ï€/3) + i sin(4Ï€/3)).
Write in rectangular form: 4(cos(4Ï€/3) + i sin(4Ï€/3)).
Evaluate √6(cos 90 + i sin 90) x 4(cos 120 + i sin 120).
Evaluate √6(cos 90 + i sin 90) x 4(cos 120 + i sin 120).
Evaluate √31 (cos 240 + isin 240) / √31 (cos 210 + i sin 210).
Evaluate √31 (cos 240 + isin 240) / √31 (cos 210 + i sin 210).
Find the cube of [(3√3)/2 + (3i/2)]^3.
Find the cube of [(3√3)/2 + (3i/2)]^3.
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Study Notes
Trigonometric Values and Equations
-
For ( \cos x = -\frac{4}{7} ) (where ( \sin x > 0 )):
- ( \csc x = \frac{7 \sqrt{33}}{33} )
- ( \cot x = \frac{-4 \sqrt{33}}{33} )
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When ( \tan x = \frac{4}{3} ), ( \pi \leq x \leq \frac{3\pi}{2} ):
- ( \cos 2x = -\frac{7}{25} )
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When ( \cos x = -\frac{8}{17} ), ( \pi \leq x \leq \frac{3\pi}{2} ):
- ( \cos 2x = -\frac{161}{289} )
Solving Trigonometric Equations
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Solve ( (\sqrt{2}) \sin(x) \csc(x) + 2 \csc(x) = 2 \sin(x) + 2 \csc(x) ):
- Solutions are ( x = \frac{\pi}{4}, \frac{3\pi}{4} )
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Solve ( \cos^2(x) + 2 = 2\cos(x) + 1 ):
- Solutions are ( x = 0, 2\pi )
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For ( 2 \tan(x) - \sec^2(x) = 0 ):
- Solutions are ( x = \frac{\pi}{4}, \frac{5\pi}{4} )
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Solve ( -\csc^2(x) + 2 = 0 ):
- Solutions are ( x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} )
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When ( \cos(x) = \cos(2x) ):
- Solutions include ( x = 0, \frac{2\pi}{3}, \frac{4\pi}{3} )
Angle and Vector Operations
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Solve for ( x ) in ( 5 + \sin x = \frac{10 + \sqrt{3}}{2} ):
- Solutions are ( x = \frac{\pi}{3}, \frac{2\pi}{3} )
-
For vector ( u = -6i - 3j ) and ( v = 2i - 8j ):
- Angle between vectors is ( 77.47 ) degrees
Triangle Solutions
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Solve triangle with sides ( c = 17 ) cm, ( a = 15 ) cm, ( b = 28 ) cm:
- Angles are ( A = 27.03^\circ, B = 121.96^\circ, C = 31.101^\circ )
-
Triangle with angle ( C = 65^\circ ), ( b = 35 ) yd, ( c = 32 ) yd:
- Angles ( B = 82.43^\circ, A = 32.57^\circ, a = 19.01 ) yd
-
Triangle with angle ( C = 62^\circ ), ( b = 15 ) mi, ( c = 10 ) mi:
- No valid triangle (SSA condition)
Vector Operations and Properties
-
For ( u ) and ( g ):
- Component form magnitude ( 2\sqrt{41} ) at direction angle ( 128.66^\circ )
-
For the dot product of ( u = -8i - 5j ) and ( v = 7i - 4j ):
- Resulting dot product is ( -36 )
-
Check vectors ( u ) and ( v ) for orthogonality or parallelism:
- Some are orthogonal while others are neither
Complex Numbers and Polar Coordinates
-
To express ( -\sqrt{3} + i ) in trigonometric form:
- Equivalent to ( 2(\cos(-\frac{\pi}{6}) + i \sin(-\frac{\pi}{6})) )
-
Convert ( 4(\cos(60) + i\sin(60)) ) to rectangular form:
- Result is ( 2 + 2i\sqrt{3} )
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Compute the product ( \sqrt{6}(\cos(90) + i\sin(90)) \times 4(\cos(120) + i\sin(120)) ):
- Resulting in ( -6\sqrt{2} - 2\sqrt{6} i )
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