Practice sheet

Fresh questions from the whole chapter, to do on paper.

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Convert $225^\circ$ to radians. Give the exact value.
Convert $\pi$ radians to degrees.
A circle has radius 4 cm. What is the length of the arc cut off by a central angle of $\frac{3 \pi}{4}$ ? Give the exact value, in cm.
Find the exact value of $\cos\left(\frac{\pi}{6}\right)$ .
Find the exact value of $\sin\left(\frac{\pi}{2}\right)$ .
Match each angle with its point $(\cos\theta, \sin\theta)$ on the unit circle.
1. $\dfrac{\pi}{6}$
2. $\dfrac{\pi}{4}$
3. $\dfrac{\pi}{3}$
4. $\dfrac{\pi}{2}$
1. $\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$
2. $\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$
3. $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$
4. $(0, 1)$
Find the exact value of $\cos\left(\frac{5 \pi}{4}\right)$ .
What is the reference angle of $200^\circ$ , in degrees?
In which quadrant is $\sin\theta < 0$ and $\cos\theta > 0$ ?
1. Quadrant II
2. Quadrant III
3. Quadrant IV
$\sin\theta = \frac{1}{5}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .
Which expression is equal to $\dfrac{1 - \cos^2 x}{\sin x}$ (where $\sin x \ne 0$ )?
1. $\sin x$
2. $\cos x$
3. $1 - \cos x$
Which equation is true for every $x$ ?
1. $\sin^2 x + \cos^2 x = 1$
2. $\sin x + \cos x = 1$
3. $\sin^2 x - \cos^2 x = 1$
4. $\tan x = \dfrac{\cos x}{\sin x}$
What is the amplitude of $y = 2\sin(5x) + 1$ ?
What is the period of $y = \sin(6x)$ ?
What is the midline of $y = 9\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 9$
What is the midline of $y = 8\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 8$
What is the midline of $y = 5\cos x - 1$ ?
1. $y = -1$
2. $y = 1$
3. $y = 5$
What is the midline of $y = 4\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 4$
What is the midline of $y = 8\cos x - 7$ ?
1. $y = -7$
2. $y = 7$
3. $y = 8$
What is the midline of $y = 6\cos x - 1$ ?
1. $y = -1$
2. $y = 1$
3. $y = 6$
What is the midline of $y = 8\cos x - 2$ ?
1. $y = -2$
2. $y = 2$
3. $y = 8$
What is the midline of $y = 9\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 9$
What is the midline of $y = 6\cos x - 5$ ?
1. $y = -5$
2. $y = 5$
3. $y = 6$
What is the midline of $y = 4\cos x - 5$ ?
1. $y = -5$
2. $y = 5$
3. $y = 4$
What is the midline of $y = 2\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 2$
What is the midline of $y = 8\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 8$
What is the midline of $y = 9\cos x - 7$ ?
1. $y = -7$
2. $y = 7$
3. $y = 9$
What is the midline of $y = 6\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 6$
What is the midline of $y = 6\cos x - 8$ ?
1. $y = -8$
2. $y = 8$
3. $y = 6$
What is the midline of $y = 3\cos x - 1$ ?
1. $y = -1$
2. $y = 1$
3. $y = 3$
What is the midline of $y = 9\cos x - 2$ ?
1. $y = -2$
2. $y = 2$
3. $y = 9$
What is the midline of $y = 7\cos x - 1$ ?
1. $y = -1$
2. $y = 1$
3. $y = 7$
What is the midline of $y = 3\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 3$
What is the midline of $y = 2\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 2$
What is the midline of $y = 2\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 2$
What is the midline of $y = 5\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 5$
What is the midline of $y = 7\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 7$
What is the midline of $y = 6\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 6$
What is the midline of $y = 3\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 3$
What is the midline of $y = 5\cos x - 7$ ?
1. $y = -7$
2. $y = 7$
3. $y = 5$
What is the midline of $y = 7\cos x - 8$ ?
1. $y = -8$
2. $y = 8$
3. $y = 7$
What is the midline of $y = 5\cos x - 2$ ?
1. $y = -2$
2. $y = 2$
3. $y = 5$
What is the midline of $y = 2\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 2$
What is the midline of $y = 8\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 8$
What is the midline of $y = 7\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 7$
What is the midline of $y = 3\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 3$
What is the midline of $y = 7\cos x - 5$ ?
1. $y = -5$
2. $y = 5$
3. $y = 7$
What is the midline of $y = 5\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 5$
What is the midline of $y = 2\cos x - 1$ ?
1. $y = -1$
2. $y = 1$
3. $y = 2$
What is the midline of $y = 2\cos x - 8$ ?
1. $y = -8$
2. $y = 8$
3. $y = 2$
What is the midline of $y = 7\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 7$
What is the midline of $y = 6\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 6$
What is the midline of $y = 9\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 9$
What is the midline of $y = 3\cos x - 5$ ?
1. $y = -5$
2. $y = 5$
3. $y = 3$
What is the midline of $y = 3\cos x - 2$ ?
1. $y = -2$
2. $y = 2$
3. $y = 3$
What is the midline of $y = 9\cos x - 8$ ?
1. $y = -8$
2. $y = 8$
3. $y = 9$
What is the midline of $y = 7\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 7$
What is the midline of $y = 5\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 5$
What is the midline of $y = 4\cos x - 6$ ?
1. $y = -6$
2. $y = 6$
3. $y = 4$
What is the midline of $y = 2\cos x - 5$ ?
1. $y = -5$
2. $y = 5$
3. $y = 2$
What is the midline of $y = 5\cos x - 3$ ?
1. $y = -3$
2. $y = 3$
3. $y = 5$
What is the midline of $y = 7\cos x - 2$ ?
1. $y = -2$
2. $y = 2$
3. $y = 7$
What is the midline of $y = 9\cos x - 5$ ?
1. $y = -5$
2. $y = 5$
3. $y = 9$
What is the midline of $y = 9\cos x - 1$ ?
1. $y = -1$
2. $y = 1$
3. $y = 9$
Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 2, period $\frac{\pi}{2}$ and midline $y = 9$ .
The graph of $y = \cos(2x - 4)$ is the graph of $y = \cos(2x)$ shifted to the right. By how much?
What is the largest value of $y = 5\sin x + 4$ ?
A triangle has sides 5 cm and 12 cm with an angle of 30° between them. What is its area, in cm²?
A triangle has sides 5 m and 3 m with an angle of 60° between them. What is its exact area, in m²?
In triangle PQR, PQ = 9, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 8, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 4, PR = 13. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 8, PR = 15. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 7, PR = 10. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 9, PR = 14. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 6, PR = 14. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 5, PR = 15. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 8, PR = 10. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 9, PR = 13. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 5, PR = 10. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 6, PR = 10. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 6, PR = 12. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 7, PR = 13. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 4, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 6, PR = 15. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 6, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 7, PR = 15. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 5, PR = 14. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 5, PR = 12. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 5, PR = 13. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 6, PR = 13. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 8, PR = 12. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 7, PR = 14. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 7, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 9, PR = 10. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 5, PR = 11. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 9, PR = 12. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 4, PR = 10. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 7, PR = 12. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 4, PR = 14. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 8, PR = 14. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 4, PR = 12. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 9, PR = 15. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 8, PR = 13. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle PQR, PQ = 4, PR = 15. To find the area with ½ · PQ · PR · sin(angle), which angle do you use?
1. The angle at P, between the two sides
2. The angle at Q
3. Any angle of the triangle
In triangle ABC, angle A = 30°, angle B = 45° and side a = 2 (opposite A). Use the law of sines to find side b (opposite B), exactly.
In triangle ABC, angle A = 30°, side a = 7 and side b = 11. Use the law of sines to find sin B.
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 7
2. When you know the three sides
3. When you know two sides and the angle between them
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 3
2. When you know the three sides
3. When you know two sides and the angle between them
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 4
2. When you know the three sides
3. When you know two sides and the angle between them
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 8
2. When you know the three sides
3. When you know two sides and the angle between them
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 6
2. When you know the three sides
3. When you know two sides and the angle between them
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 9
2. When you know the three sides
3. When you know two sides and the angle between them
You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
1. When you know two angles and a side, such as A = 40°, B = 65° and a = 5
2. When you know the three sides
3. When you know two sides and the angle between them
In triangle ABC, a = 12, b = 8 and the angle between them is C = 60°. Use the law of cosines to find c².
In triangle ABC, a = 12, b = 5 and C = 120°. Use the law of cosines to find c².
A triangle has sides a = 5, b = 6 and c = 8. Find cos C, the cosine of the angle opposite c.

Answers

$\frac{5 \pi}{4}$
$180$
$3 \pi$
$\frac{\sqrt{3}}{2}$
$1$
$\dfrac{\pi}{6}$ → $\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$ · $\dfrac{\pi}{4}$ → $\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$ · $\dfrac{\pi}{3}$ → $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$ · $\dfrac{\pi}{2}$ → $(0, 1)$
$- \frac{\sqrt{2}}{2}$
$20$
Quadrant IV
$- \frac{2 \sqrt{6}}{5}$
$\sin x$
$\sin^2 x + \cos^2 x = 1$
$2$
$\frac{\pi}{3}$
$y = -4$
$2 \sin{\left(4 x \right)} + 9$
$2$
$9$
$15$
$\frac{15 \sqrt{3}}{4}$
The angle at P, between the two sides
$2 \sqrt{2}$
$\frac{11}{14}$
When you know two angles and a side, such as A = 40°, B = 65° and a = 7
$112$
$229$
$- \frac{1}{20}$