Practice
Trigonometric Functions of General Angles and the Unit Circle Practice
Fifty original questions on coordinate definitions, exact unit-circle values, reference angles, quadrants, axes, and synthesis.
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Question 1
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(\frac{15}{17}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 2
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(-\frac8{17}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 3
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(-\frac{15}{8}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 4
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(-\frac{21}{29}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 5
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(\frac{20}{29}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 6
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(-\frac{21}{20}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 7
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(-\frac{35}{37}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 8
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(-\frac{12}{37}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 9
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(\frac{40}{9}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 10
Explanation
Use \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\), with reciprocals as needed. Substitution gives \(\frac{61}{11}\).
- Reasoning
Read the coordinate signs first, then divide by the radius or the other coordinate.
Question 11
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac12\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 12
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac{\sqrt3}{2}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 13
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac{\sqrt3}{3}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 14
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac{\sqrt2}{2}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 15
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac{\sqrt2}{2}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 16
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(1\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 17
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac{\sqrt3}{2}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 18
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac12\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 19
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\sqrt3\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 20
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac{\sqrt3}{2}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 21
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(-\frac{\sqrt2}{2}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 22
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(-\frac{\sqrt3}{3}\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 23
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(-\frac12\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 24
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(\frac12\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 25
Explanation
Use the familiar reference angle and the terminal quadrant. The exact value is \(-1\).
- Reasoning
Use a special-triangle magnitude, then apply the coordinate sign from the quadrant.
Question 26
Explanation
In quadrant II, subtract from 180°: 180°−140°=40°.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 27
Explanation
In quadrant III, subtract 180°: 220°−180°=40°.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 28
Explanation
In quadrant IV, subtract from 360°: 360°−310°=50°.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 29
Explanation
In quadrant II, y is positive and x is negative, so sine is positive, cosine negative, and tangent negative.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 30
Explanation
Both coordinates are negative in quadrant III, making sine and cosine negative but their quotient positive.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 31
Explanation
In quadrant IV, y is negative and x positive, so tangent is negative.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 32
Explanation
At π/2 the terminal side meets the top of the circle at (0,1).
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 33
Explanation
At π the terminal side meets the negative x-axis at (−1,0).
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 34
Explanation
At π/2, x=cos θ=0, so y/x divides by zero and tangent is undefined.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 35
Explanation
At 3π/2, cosine is zero, so tangent is undefined.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 36
Explanation
The reference angle is π/6 and sine is negative in quadrant III.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 37
Explanation
The reference angle is π/4 and cosine is negative in quadrant III.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 38
Explanation
The reference angle is π/3 and tangent is negative in quadrant IV.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 39
Explanation
A unit circle has center at the origin and radius 1, so x²+y²=1.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 40
Explanation
Negative x and positive y place the point in quadrant II; the coordinates have a π/6 reference angle, giving 5π/6.
- Reasoning
Locate the terminal point, find the acute reference angle when needed, and use coordinate signs.
Question 41
Explanation
The radius is 29, so sine is 21/29 and cosine is −20/29; their sum is 1/29.
- Reasoning
Compute the radius, preserve coordinate signs, then combine exact fractions.
Question 42
Explanation
A 3-4-5 triangle gives cosine 4/5 in quadrant IV, so tangent=(−3/5)/(4/5)=−3/4.
- Reasoning
Use the quadrant to choose the sign of the missing coordinate ratio.
Question 43
Explanation
The conditions place θ in quadrant II. The 5-12-13 relationship gives positive sine 5/13, so cosecant is 13/5.
- Reasoning
Use the stated sign to locate the quadrant before taking a square root.
Question 44
Explanation
Reduce 13π/6 to π/6 and 7π/3 to π/3. The values are 1/2 and 1/2, summing to 1.
- Reasoning
Reduce each angle separately by 2π before evaluating.
Question 45
Explanation
Tangent has period π, so tan(17π/4)=tan(π/4)=1. Sine(11π/6)=−1/2; subtracting gives 3/2.
- Reasoning
Use each function's own period, then apply exact signs.
Question 46
Explanation
Substitute y=x into x²+y²=1: 2x²=1. Quadrant III requires the negative root, x=−√2/2.
- Reasoning
Use the circle equation, then choose the root from the quadrant.
Question 47
Explanation
The radius is 10 and y/r=4/5, so y=8. Then x²=100−64=36; x<0 gives x=−6.
- Reasoning
Translate the ratio into a coordinate before using the circle equation.
Question 48
Explanation
Tangent is y/x=(−a)/a=−1; quadrant IV confirms the negative sign.
- Reasoning
Use y/x directly when both coordinates are expressed with the same parameter.
Question 49
Explanation
sin 150°=1/2, so its square is 1/4. tan 315°=−1, so its square is 1; total 5/4.
- Reasoning
Evaluate exact values before applying exponents.
Question 50
Explanation
The radius is \(25\). Thus \(\sec\theta=-25/7\) and \(\csc\theta=-25/24\), so \(\sec\theta-\csc\theta=-25/7+25/24=-425/168\).
- Reasoning
Compute reciprocal ratios with signs, then combine using a common denominator.
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