Trigonometry in solids

Trigonometry, grade 11 (5 units) Lesson 16 of 18Learning

Trigonometry in solids

The angle between a line and a plane

A right solid – a right prism, a right pyramid, a cylinder, a cone – has a height perpendicular to the base. So every triangle with the height as one of its sides is right-angled, and is solved with $\displaystyle \sin$ , $\displaystyle \cos$ and $\displaystyle \tan$ .

The angle between a line and a plane: the angle between the line and its projection on the plane. The projection of $\displaystyle SA$ on the base is $\displaystyle OA$ , where $\displaystyle O$ is the foot of the height from $\displaystyle S$ .
The angle between a face and the base: the angle between two lines perpendicular to their common edge – one in the face and one in the base. In a right pyramid on a square base: $\displaystyle \angle SMO$ , where $\displaystyle M$ is the midpoint of a base edge.

A right square pyramid

$\displaystyle SABCD$ is a right pyramid on a square base $\displaystyle ABCD$ of side $\displaystyle 6$ . $\displaystyle O$ is the centre of the base.
The lateral edge $\displaystyle SA$ makes an angle of $\displaystyle 50^\circ$ with the base.
Find the height, $\displaystyle SA$ and the volume of the pyramid.

$\displaystyle OA$ is half the diagonal: $\displaystyle {OA}={\frac{6\sqrt{2}}{2}}={3\sqrt{2}}$ .
In the right-angled triangle $\displaystyle SOA$ : $\displaystyle {SO}={3\sqrt{2}\tan50^\circ\approx5.06}$ , and $\displaystyle {SA}={\frac{3\sqrt{2}}{\cos50^\circ}\approx6.60}$ .
$\displaystyle {V}={\frac{S\cdot h}{3}}={\frac{36\cdot3\sqrt{2}\tan50^\circ}{3}\approx60.67}$ .

A cone

The slant height of a right cone is $\displaystyle 10$ cm, and it makes an angle of $\displaystyle 35^\circ$ with the base.
Find the radius, the height and the volume of the cone.

In triangle $\displaystyle SOA$ : $\displaystyle {r}={OA}={10\cos35^\circ\approx8.19}$ , and $\displaystyle {h}={SO}={10\sin35^\circ\approx5.74}$ .
$\displaystyle {V}={\frac{\pi r^2h}{3}}={\frac{\pi\cdot(10\cos35^\circ)^2\cdot10\sin35^\circ}{3}\approx403.04}$ cm³.
Round only at the end: with $\displaystyle 8.19$ and $\displaystyle 5.74$ it comes out $\displaystyle 403.19$ .

Exercise 1

$\displaystyle ABCDA'B'C'D'$ is a box: $\displaystyle {AB}={4}$ , $\displaystyle {BC}={3}$ .
The diagonal $\displaystyle AC'$ makes an angle of $\displaystyle 40^\circ$ with the base $\displaystyle ABCD$ .

a

Find the height of the box,

\displaystyle CC'

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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b

Find the volume of the box.

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Exercise 2

$\displaystyle SABCD$ is a right pyramid on a square base of side $\displaystyle 8$ .
A lateral face makes an angle of $\displaystyle 60^\circ$ with the base.

a

Find the height of the pyramid. A hint: the triangle

\displaystyle SOM

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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b

Find the area of the four lateral faces together.

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Exercise 3

The base radius of a right cylinder is $\displaystyle 5$ .
The diagonal of the axial section (the rectangle through the cylinder’s axis) makes an angle of $\displaystyle 30^\circ$ with the base.
Find the volume of the cylinder.

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Exercise 4

A plane cuts a sphere of radius $\displaystyle 13$ , at a distance $\displaystyle 5$ from the sphere’s centre $\displaystyle O$ . The section is a circle.
$\displaystyle A$ is a point on the section’s circle.
Find the radius of the section’s circle, and the angle between $\displaystyle OA$ and the plane of the section.

The section’s radius

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The angle between

\displaystyle OA

and the plane

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Exercise 5

The base of a right prism is an isosceles triangle: the legs are $\displaystyle 6$ , and the apex angle is $\displaystyle 40^\circ$ .
The height of the prism is $\displaystyle 10$ .
Find the volume of the prism.

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Formulas: trigonometry, grade 11 (5 units)

Basic identities

The Pythagorean identity
$\displaystyle \sin^{2}\alpha+\cos^{2}\alpha$ $\displaystyle =$ $\displaystyle 1$
The tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$

The sum and the difference of angles

$\displaystyle \cos(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \cos\alpha\cdot\cos\beta\mp\sin\alpha\cdot\sin\beta$
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$
$\displaystyle \tan(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\cdot\tan\beta}$

The double angle

$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
$\displaystyle \cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha=2\cos^{2}\alpha-1=1-2\sin^{2}\alpha$
$\displaystyle \tan2\alpha$ $\displaystyle =$ $\displaystyle \frac{2\tan\alpha}{1-\tan^{2}\alpha}$
$\displaystyle \cos^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1+\cos2\alpha}{2}$
$\displaystyle \sin^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1-\cos2\alpha}{2}$

Sums and differences of sines and of cosines

$\displaystyle \sin\alpha+\sin\beta$ $\displaystyle =$ $\displaystyle 2\sin\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}$
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\sin\frac{\alpha-\beta}{2}\cdot\cos\frac{\alpha+\beta}{2}$
$\displaystyle \cos\alpha+\cos\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}$
$\displaystyle \cos\alpha-\cos\beta$ $\displaystyle =$ $\displaystyle -2\sin\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$

Equations – the general solution ($k$ a whole number)

$\displaystyle \sin x=\sin\alpha\iff x=\alpha+2\pi k\ \text{or}\ x=\pi-\alpha+2\pi k$
$\displaystyle \cos x=\cos\alpha\iff x=\pm\alpha+2\pi k$
$\displaystyle \tan x=\tan\alpha\iff x=\alpha+\pi k$

A triangle

The law of sines ( $\displaystyle R$ – the radius of the circumscribed circle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^{2}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}-2a\cdot b\cdot\cos\gamma$
The area of a triangle ( $\displaystyle \alpha$ – the angle between $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}b\cdot c\cdot\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ , $\displaystyle \gamma$ – the angles next to $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^{2}\cdot\sin\beta\cdot\sin\gamma}{2\cdot\sin(\beta+\gamma)}$

A circle ($\alpha$ in radians, $R$ – the radius)

The area of a disc
$\displaystyle S$ $\displaystyle =$ $\displaystyle \pi R^{2}$
The circumference of a circle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\pi R$
The length of an arc
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha\cdot R$
The area of a sector
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha\cdot R^{2}$

Solids ($S$ – the base area, $h$ – the height, $l$ – the slant height)

The volume of a prism and a cylinder
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\cdot h$
The volume of a pyramid and a cone
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\cdot h}{3}$
The lateral surface of a cone
$\displaystyle M$ $\displaystyle =$ $\displaystyle \pi R\cdot l$
The volume of a sphere
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4}{3}\pi R^{3}$
The surface area of a sphere
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^{2}$