Questions and Answers
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.
-7/25
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).
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Solve: cos^2(x) + 2 = 2cos(x) + 1.
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What are the solutions to the equation 0 = 2 tan(x) - sec^2(x)?
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Solve -csc^2(x) + 2 = 0.
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If cos(x) = cos 2(x), what are the solutions?
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Solve for x in the equation 5 + sin x = (10 + √3)/2.
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Find x such that arctan(tan(3π/4)) = x.
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What is x if arctan(sec(π)) = x?
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Find x if arccos(sin(0)) = x.
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What is x if arcsin(csc(π/2)) = x?
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Solve the triangle with c = 17 cm, a = 15 cm, b = 28 cm.
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Solve the triangle where angle C = 65 degrees, b = 35 yd, c = 32 yd.
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Solve the triangle with angle C = 62 degrees, b = 15 mi, c = 10 mi.
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What is cos(-7π/12)?
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Find tan(5π/12).
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Given u = and v = , find -u - v.
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Given u = and g = , find u + g.
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Find the angle between the two vectors u = -6i - 3j and v = 2i - 8j.
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Find the angle between the vectors u = and v = .
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Write the vector CD in component form where C = and D = .
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Find the component form of the resultant vector if a = and find 2a.
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What is the direction angle for AB where A = and B = ?
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Find the dot product of the vectors u = and v = .
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Find the dot product of u = -8i - 5j and v = 7i - 4j.
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State if the vector u = and v = is parallel, orthogonal, or neither.
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State if the vector u = and v = is parallel, orthogonal, or neither.
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Write the vector -√3 + i in trig form.
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Write the vector √6 - 3i√2 in trig form.
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Write in rectangular form: 4(cos 60 + i sin 60).
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Write in rectangular form: 4(cos(4π/3) + i sin(4π/3)).
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Evaluate √6(cos 90 + i sin 90) x 4(cos 120 + i sin 120).
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Evaluate √31 (cos 240 + isin 240) / √31 (cos 210 + i sin 210).
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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} )
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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 )
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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
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Triangle with angle ( C = 62^\circ ), ( b = 15 ) mi, ( c = 10 ) mi:
- No valid triangle (SSA condition)
Vector Operations and Properties
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For ( u ) and ( g ):
- Component form magnitude ( 2\sqrt{41} ) at direction angle ( 128.66^\circ )
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For the dot product of ( u = -8i - 5j ) and ( v = 7i - 4j ):
- Resulting dot product is ( -36 )
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Check vectors ( u ) and ( v ) for orthogonality or parallelism:
- Some are orthogonal while others are neither
Complex Numbers and Polar Coordinates
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To express ( -\sqrt{3} + i ) in trigonometric form:
- Equivalent to ( 2(\cos(-\frac{\pi}{6}) + i \sin(-\frac{\pi}{6})) )
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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 )
Studying That Suits You
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Description
Test your knowledge of precalculus with these flashcards. This quiz covers topics such as csc, cot, and trigonometric identities based on different values of x. Perfect for final preparations in precalculus.
