One question for each chapter

Do the problems in more than one session, without rushing. First draw and explain which tool you chose. Check the resolutions after the attempt. If a topic still seems new, go back to the chapter indicated by the challenge number.

1. Rotates. A wheel with radius 12 cm rotates $120°$. How far does an edge point travel along the circumference?

Resolution 1

$120°=2\pi/3$ rad. The arc length is $s=r\theta=12\cdot2\pi/3=8\pi$ cm. It is one third of the total length $24\pi$ cm.

2. Waves. The height of a cabin is $h(\theta)=9+7\operatorname{sen}\theta$, in meters. What are the minimum and maximum heights?

Resolution 2

As $-1\le\operatorname{sen}\theta\le1$, the minimum height is $9-7=2$ m and the maximum is $9+7=16$ m. The amplitude is 7 m, half the difference between the extreme heights.

3. Identities. If $\cos\theta=-7/25$ and $\theta$ is in the second quadrant, find sine and tangent.

Resolution 3

$\operatorname{sen}^2\theta=1-49/625=576/625$. The sine is positive in the second quadrant: $\operatorname{sen}\theta=24/25$. The tangent is $(24/25)/(-7/25)=-24/7$.

4. Symmetry. Calculate $\operatorname{sen}(-150°)$ and $\cos240°$.

Resolution 4

Since sine is odd, $\operatorname{sen}(-150°)=-\operatorname{sen}150°=-1/2$. Such as $240°=180°+60°$, $\cos240°=-\cos60°=-1/2$.

5. Polygons. Calculate the side and perimeter of a equilateral triangle inscribed in a circle of radius 4 cm.

Resolution 5

For $n=3$, $\ell=2R\operatorname{sen}(\pi/3)=8\cdot\sqrt3/2=4\sqrt3$ cm. The perimeter is $3\ell=12\sqrt3$ cm.

6. Transformations. Find $\tan75°$ using $75°=45°+30°$.

Resolution 6

Using the sum formula:

$$\tan75°=\frac{1+1/\sqrt3}{1-1/\sqrt3} =\frac{\sqrt3+1}{\sqrt3-1}=2+\sqrt3.$$

In the last step, we multiply by $(\sqrt3+1)/(\sqrt3+1)$. No denominator is zero.

7. Equations. Solve $2\operatorname{sen} x-\sqrt3=0$ into $[0°,360°)$.

Resolution 7

$\operatorname{sen} x=\sqrt3/2$. The reference angle is $60°$. The sine is positive in the first and second quadrants: $x=60°$ or $x=120°$.

8. Inequalities. Solve $\cos x\le-1/2$ into $[0°,360°)$.

Resolution 8

Equalities occur in $120°$ and $240°$. The arc to the left of the vertical line with coordinate $-1/2$ is between these angles. Since inequality includes equality, the solution is $[120°,240°]$.

9. A measurement in the courtyard. Your eyes are 1,2 m from the ground. At 10 m horizontal distance from a pole, on flat ground, you measure $30°$ elevation to the top. How tall is the post?

Resolution 9

The difference in level between the eyes and the top is $10\tan30°=10\sqrt3/3$ m. The total height is $1{,}2+10\sqrt3/3$ m, approximately 6,97 m.

10. A complete triangle. A right-angled triangle has legs 6 and 8 cm and hypotenuse 10 cm. Calculate area, radius of the circumscribed circle and radius of the inscribed circle.

Resolution 10

The area is $K=6\cdot8/2=24$ cm². Using the formula $K=abc/(4R)$, we have $R=(6\cdot8\cdot10)/(4\cdot24)=5$ cm. The semiperimeter is $s=(6+8+10)/2=12$ cm, then $r=K/s=2$ cm.

Closing project

Make a page with a figure of your own, a demonstration that you can explain aloud and a new problem about rotations or measurements. Solve your problem before passing it on to someone else. The explanation must make the data, the unit of the angle and the reason for choosing the formula clear.

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