Formula of inverse trigonometric functions
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A problem: In the triangle below, what is the measure of angle LLL?
LL35356565??
What we know: Relative to \angle L∠Langle, L, we know the lengths of the opposite and adjacent sides, so we can write:
\tan(L) = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{35}{65}tan(L)=adjacentopposite=6535tangent, left parenthesis, L, right parenthesis, equals, start fraction, o, p, p, o, s, i, t, e, divided by, a, d, j, a, c, e, n, t, end fraction, equals, start fraction, 35, divided by, 65, end fraction
But this doesn't help us find the measure of \angle L∠Langle, L. We're stuck!
What we need: We need new mathematical tools to solve problems like these. Our old friends sine, cosine, and tangent aren’t up to the task. They take angles and give side ratios, but we need functions that take side ratios and give angles. We need inverse trig functions!
The inverse trigonometric functions
We already know about inverse operations. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. Each operation does the opposite of its inverse.
The idea is the same in trigonometry. Inverse trig functions do the opposite of the “regular” trig functions. For example:
Inverse sine (\sin^{-1})(sin−1)left parenthesis, sine, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the sine.Inverse cosine (\cos^{-1})(cos−1)left parenthesis, cosine, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the cosine.Inverse tangent (\tan^{-1})(tan−1)left parenthesis, tangent, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the tangent.
In general, if you know the trig ratio but not the angle, you can use the corresponding inverse trig function to find the angle. This is expressed mathematically in the statements below.
Trigonometric functions input angles and output side ratiosInverse trigonometric functions input side ratios and output angles\sin (\theta)=\dfrac {\text{opposite}}{\text{hypotenuse}}sin(θ)=hypotenuseoppositesine, left parenthesis, theta, right parenthesis, equals, start fraction, o, p, p, o, s, i, t, e, divided by, h, y, p, o, t, e, n, u, s, e, end fraction\rightarrow→right arrow\sin^{-1}\left(\dfrac {\text{opposite}}{\text{hypotenuse}}\right)=\thetasin−1(hypotenuse
LL35356565??
What we know: Relative to \angle L∠Langle, L, we know the lengths of the opposite and adjacent sides, so we can write:
\tan(L) = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{35}{65}tan(L)=adjacentopposite=6535tangent, left parenthesis, L, right parenthesis, equals, start fraction, o, p, p, o, s, i, t, e, divided by, a, d, j, a, c, e, n, t, end fraction, equals, start fraction, 35, divided by, 65, end fraction
But this doesn't help us find the measure of \angle L∠Langle, L. We're stuck!
What we need: We need new mathematical tools to solve problems like these. Our old friends sine, cosine, and tangent aren’t up to the task. They take angles and give side ratios, but we need functions that take side ratios and give angles. We need inverse trig functions!
The inverse trigonometric functions
We already know about inverse operations. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. Each operation does the opposite of its inverse.
The idea is the same in trigonometry. Inverse trig functions do the opposite of the “regular” trig functions. For example:
Inverse sine (\sin^{-1})(sin−1)left parenthesis, sine, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the sine.Inverse cosine (\cos^{-1})(cos−1)left parenthesis, cosine, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the cosine.Inverse tangent (\tan^{-1})(tan−1)left parenthesis, tangent, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the tangent.
In general, if you know the trig ratio but not the angle, you can use the corresponding inverse trig function to find the angle. This is expressed mathematically in the statements below.
Trigonometric functions input angles and output side ratiosInverse trigonometric functions input side ratios and output angles\sin (\theta)=\dfrac {\text{opposite}}{\text{hypotenuse}}sin(θ)=hypotenuseoppositesine, left parenthesis, theta, right parenthesis, equals, start fraction, o, p, p, o, s, i, t, e, divided by, h, y, p, o, t, e, n, u, s, e, end fraction\rightarrow→right arrow\sin^{-1}\left(\dfrac {\text{opposite}}{\text{hypotenuse}}\right)=\thetasin−1(hypotenuse
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