Transforming Tangents?

Geometry Level 3

$\large \tan70^\circ - \tan50^\circ + \tan10^\circ$

The expression above has a closed form, find this closed form.
Give your answer to 2 decimal places.

The answer is 1.73.

1 solution

Arkajyoti Banerjee
Jul 25, 2016

Relevant wiki: Sum and Difference Trigonometric Formulas - Problem Solving

$a=\tan 70^{\circ} - \tan 50^{\circ} + \tan 10^{\circ}$

= $\tan (60+10)^{\circ} - \tan (60-10)^{\circ} + \tan 10^{\circ}$

= $\frac{\tan 60 +\tan 10}{1-\tan 60\tan 10} + \frac{\tan 10-\tan 60}{1+\tan 60\tan 10} + \tan 10$

= $\frac{(\sqrt{3}+\tan 10)(1+\sqrt{3}\tan 10)-(\sqrt{3}-\tan 10)(1-\sqrt{3}\tan 10)}{(1-\sqrt{3}\tan 10)(1+\sqrt{3}\tan 10)}+\tan 10$

= $\frac{8\tan 10}{1-3\tan^{2}10}+\tan 10$

= $\frac{8\tan 10+\tan 10-3tan^{3}10}{1-3\tan^{2}10}$

= $3\frac{3\tan 10-tan^{3}10}{1-3\tan^{2}10}$

= $3 \times \tan 30$

= $3 \times \frac{1}{\sqrt{3}}$

= $\sqrt{3} \approx \boxed{1.73}$

Hi Mr. Banerjee its 1/a=0.57 and not a=0.57

NIKHESH KUMAR - 4 years, 10 months ago

Thanks. I've made the correction.

Arkajyoti Banerjee - 4 years, 10 months ago

Why are lines 7 and 8 the same? Also, you had to use a calculator to know tan60°=√3. If you could use a calculator why go through all this trouble instead of calculating the original expression?

Peleg Tsadok - 4 years, 10 months ago

tan (60) = sqrt(3) is through special angle triangles: angles 30-60-90, has respectively opposite side lengths of 1/2 - sqrt(3)/2 - 1 (derived from unit circle)

W A - 4 years, 10 months ago

What you're saying is that you memorized the value of tan(60°).

You could have just as easily memorized tan(70°), tan(50°) and tan(10°).

Peleg Tsadok - 4 years, 10 months ago

@Peleg Tsadok – We can evaluate the tangent of $60^{\circ}$ using some basic geometry, and it doesn't require any calculator. But not that of $70^{\circ}$ or $50^{\circ}$

Arkajyoti Banerjee - 4 years, 10 months ago

$30^{\circ}$ and $60^{\circ}$ are standard angles, and their trigonometric values can easily be found out by constructing a 30-60-90 triangle and applying some basic geometry. Evidently that doesn't involve the use of calculators. And everybody knows the approx value of $\sqrt{2}$ and $\sqrt{3}$ correct to 3 decimal places. And even if you don't, then you can simply use the long division method to evaluate the square root.

Arkajyoti Banerjee - 4 years, 10 months ago

That's interesting, I never knew it's possible to calculate the value of tan(60°) geometrically, can you explain that to me?

Peleg Tsadok - 4 years, 10 months ago

@Peleg Tsadok –

Construct an equilateral triangle of side $a$ . Now drop a perpendicular from $A$ to $BC$ at $D$ . Recall that the perpendicular dropped from the point opposite to the unequal side of an isosceles triangle (an equilateral triangle is also an isosceles triangle) bisects the opposite side as well as the angle. Hence we have $DC=\frac{a}{2}$ . From the Pythagorean theorem, we have $AD=\frac{\sqrt{3}}{2}a$ Hence, $\tan 60^{\circ} = \frac{AD}{DC} = \sqrt{3}$ .

Similarly, we can evaluate the trig ratios of $30^{\circ}$ too from that right triangle.

Arkajyoti Banerjee - 4 years, 10 months ago

@Arkajyoti Banerjee – Thanks!

Apparently you can't thank a person in Brilliant with only one word.

"Please provide a more complete explanation of your response before submitting a reply."

So here is a complete explanation of the concept of gratitude(copied from Wikipedia):

Gratitude, thankfulness, gratefulness, or appreciation is a feeling or attitude in acknowledgment of a benefit that one has received or will receive. The experience of gratitude has historically been a focus of several world religions, and has been considered extensively by moral philosophers such as Lee Clement. The systematic study of gratitude within psychology only began around the year 2000, possibly because psychology has traditionally been focused more on understanding distress rather than understanding positive emotions. The study of gratitude within psychology has focused on the understanding of the short term experience of the emotion of gratitude (state gratitude), individual differences in how frequently people feel gratitude (trait gratitude), and the relationship between these two aspects.

Peleg Tsadok - 4 years, 10 months ago

Transforming Tangents?

The answer is 1.73.

1 solution

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