How do you measure the height of a building without climbing it, or the gradient of a ramp without laying a ruler on it? The answer lies in right-angled triangles: one angle and one side are enough to find out everything else. This is the first time you are meeting this tool, so let's take it step by step.
Draw several right-angled triangles with the same acute angle $\alpha$ but different sizes. They are all similar (same angles), so the ratios between their corresponding sides are always the same number, however large or small the triangle is.
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Look at the right-angled triangle in the figure. a) Write, as fractions, the ratios $\sen\ \alpha$, $\cos\ \alpha$ and $\tg\ \alpha$ of the marked angle. b) Write these…
Look at the right-angled triangle in the figure.
a) Write, as fractions, the ratios $\sen\ \alpha$, $\cos\ \alpha$ and $\tg\ \alpha$ of the marked angle. b) Write these same ratios for the other acute angle. What relationship do you observe between them?
In a right-angled triangle, fix an acute angle $\alpha$. Each side has a name according to its position relative to $\alpha$: the hypotenuse, the opposite side (the one that does not touch $\alpha$) and the adjacent side (the one that does touch it, other than the hypotenuse).
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In the right-angled triangle in the figure, the legs measure $6$ cm and $8$ cm. a) Calculate the hypotenuse. b) Find the three trigonometric ratios of the marked angle…
In the right-angled triangle in the figure, the legs measure $6$ cm and $8$ cm.
a) Calculate the hypotenuse. b) Find the three trigonometric ratios of the marked angle $\beta$.
In the triangle in the figure, the hypotenuse measures $10$ cm and angle $\alpha$ is $30^\circ$. Calculate the length of the side opposite $\alpha$. (Given:…
In the triangle in the figure, the hypotenuse measures $10$ cm and angle $\alpha$ is $30^\circ$.
Calculate the length of the side opposite $\alpha$. (Given: $\sen\ 30^\circ = 0{,}5$.)
The two shorter sides of a right-angled triangle each measure $4$ cm. a) What is the tangent of each of the acute angles? b) What are the measurements of these angles?…
The two shorter sides of a right-angled triangle each measure $4$ cm. a) What is the tangent of each of the acute angles? b) What are the measurements of these angles? Justify your answer without a calculator.
Since sine and cosine come from the same right-angled triangle, they are not independent: they are linked by Pythagoras.
These relationships allow you to calculate all three ratios knowing only one of them, without measuring any side.
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Determine whether these statements about an acute angle $\alpha$ are true or false: a) "The sine of $\alpha$ can be greater than $1$." b) "The tangent of $\alpha$ can be…
Determine whether these statements about an acute angle $\alpha$ are true or false: a) "The sine of $\alpha$ can be greater than $1$." b) "The tangent of $\alpha$ can be greater than $1$." Justify both answers using the definition of the ratios in a right-angled triangle and give an example where applicable.
There are three angles whose ratios can be calculated exactly, without a calculator, from very simple shapes.
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The acute angles of a right-angled triangle are $35^\circ$ and $55^\circ$. Knowing that $\sen\ 35^\circ \approx 0{,}574$, find without a calculator the value of…
The acute angles of a right-angled triangle are $35^\circ$ and $55^\circ$. Knowing that $\sen\ 35^\circ \approx 0{,}574$, find without a calculator the value of $\cos\ 55^\circ$ and justify it using the definition of trigonometric ratios.
a) Look at Panel A in the figure, where an equilateral triangle of side $2$ has been divided by an altitude, and deduce the exact values of $\sen 30^\circ$,…
a) Look at Panel A in the figure, where an equilateral triangle of side $2$ has been divided by an altitude, and deduce the exact values of $\sen 30^\circ$, $\cos 30^\circ$, and $\tg 30^\circ$.
b) Look at Panel B, where a square of side $1$ has been divided by a diagonal, and deduce the trigonometric ratios of $45^\circ$ ($\sen 45^\circ$, $\cos 45^\circ$, and $\tg 45^\circ$).
c) Using the above and the relationships between complementary angles, complete the following table with the ratios of $30^\circ$, $45^\circ$, and $60^\circ$:
Calculate without a calculator, using the table of special angles: a) $\sen^2\ 30^\circ + \cos^2\ 60^\circ + \tg\ 45^\circ$. b)…
Calculate without a calculator, using the table of special angles: a) $\sen^2\ 30^\circ + \cos^2\ 60^\circ + \tg\ 45^\circ$. b) $\frac{\sen\ 60^\circ}{\cos\ 30^\circ} + \tg^2\ 30^\circ$.
Back to the beginning: To measure the height of the building, all you need to do is stand a known distance from its base (the adjacent side) and measure the angle of elevation $\alpha$ to the top: the height is that distance multiplied by $\tan\ \alpha$.
Summary
Let's practice! 10 medium exercises in this section