Inverse trigonometric functions formulas

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3 Polynomial and Rational Functions • Introduction to Polynomial and Rational Functions • 3.1 Complex Numbers • 3.2 Quadratic Functions • 3.3 Power Functions and Polynomial Functions • 3.4 Graphs of Polynomial Functions • 3.5 Dividing Polynomials • 3.6 Zeros of Polynomial Functions • 3.7 Rational Functions • 3.8 Inverses and Radical Functions • 3.9 Modeling Using Variation • 4 Exponential and Logarithmic Functions • Introduction to Exponential and Logarithmic Functions • 4.1 Exponential Functions • 4.2 Graphs of Exponential Functions • 4.3 Logarithmic Functions • 4.4 Graphs of Logarithmic Functions • 4.5 Logarithmic Properties • 4.6 Exponential and Logarithmic Equations • 4.7 Exponential and Logarithmic Models • 4.8 Fitting Exponential Models to Data • 7 Trigonometric Identities and Equations • Introduction to Trigonometric Identities and Equations • 7.1 Solving Trigonometric Equations with Identities • 7.2 Sum and Difference Identities • 7.3 Double-Angle, Half-Angle, and Reduction Formulas • 7.4 Sum-to-Product and Product-to-Sum Formulas • 7.5 Solving Trigonometric Equations • 7.6 Modeling with Trigonometric Functions • 8 Further Applications of Trigonometry • Introduction to Further Applications of Trigonometry • 8.1 Non-right Triangles: Law of Sines • 8.2 Non-right Triangles: Law of Cosines • 8.3 Polar Coordinates • 8.4 Polar Coordinates: Graphs • 8.5 Polar Form of Complex Numbers • 8.6 Parametric Equations • 8.7 Parametric Equations: Graphs • 8.8 Vectors • 9 Systems of...

tan ⁡ ( L ) = opposite adjacent = 35 65 \tan(L) = \dfrac tan ( L ) = adjacent opposite ​ = 6 5 3 5 ​ tangent, left parenthesis, L, right parenthesis, equals, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, a, d, j, a, c, e, n, t, end text, end fraction, equals, start fraction, 35, divided by, 65, end fraction • Inverse sine ( sin ⁡ − 1 ) (\sin^) ( tan − 1 ) left parenthesis, tangent, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the tangent. Trigonometric functions input angles and output side ratios Inverse trigonometric functions input side ratios and output angles sin ⁡ ( θ ) = opposite hypotenuse \sin (\theta)=\dfrac \right)=\theta tan − 1 ( adjacent opposite ​ ) = θ tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, a, d, j, a, c, e, n, t, end text, end fraction, right parenthesis, equals, theta The expression sin ⁡ − 1 ( x ) \sin^ sin ( x ) 1 ​ start fraction, 1, divided by, sine, left parenthesis, x, right parenthesis, end fraction . In other words, the − 1 -1 − 1 minus, 1 is not an exponent. Instead, it simply means inverse function. A coordinate plane. The x-axis starts at zero and goes to ninety by tens. It is labeled degrees. The y-axis starts at zero and goes to two by two tenths. It is labeled a ratio. The graphed line is labeled sine of x, which is a nonlinear curve. The line f...

arcsin ⁡ ( x ) \arcsin(x) arcsin ( x ) \arcsin, left parenthesis, x, right parenthesis , or sin ⁡ − 1 ( x ) \sin^(x) sin − 1 ( x ) sine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis , is the inverse of sin ⁡ ( x ) \sin(x) sin ( x ) sine, left parenthesis, x, right parenthesis . arccos ⁡ ( x ) \arccos(x) arccos ( x ) \arccos, left parenthesis, x, right parenthesis , or cos ⁡ − 1 ( x ) \cos^(x) cos − 1 ( x ) cosine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis , is the inverse of cos ⁡ ( x ) \cos(x) cos ( x ) cosine, left parenthesis, x, right parenthesis . arctan ⁡ ( x ) \arctan(x) arctan ( x ) \arctan, left parenthesis, x, right parenthesis , or tan ⁡ − 1 ( x ) \tan^(x) tan − 1 ( x ) tangent, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis , is the inverse of tan ⁡ ( x ) \tan(x) tan ( x ) tangent, left parenthesis, x, right parenthesis . Radians Degrees − π 2 ≤ arcsin ⁡ ( θ ) ≤ π 2 -\dfrac − 2 π ​ < arctan ( θ ) < 2 π ​ minus, start fraction, pi, divided by, 2, end fraction, is less than, \arctan, left parenthesis, theta, right parenthesis, is less than, start fraction, pi, divided by, 2, end fraction − 9 0 ∘ < arctan ⁡ ( θ ) < 9 0 ∘ -90^\circ<\arctan(\theta)<90^\circ − 9 0 ∘ < arctan ( θ ) < 9 0 ∘ minus, 90, degrees, is less than, \arctan, left parenthesis, theta, right parenthesis, is less than, 90, degrees The trigonometric functions aren't really inverti...

In this unit, you'll explore the power and beauty of trigonometric equations and identities, which allow you to express and relate different aspects of triangles, circles, and waves. You'll learn how to use trigonometric functions, their inverses, and various identities to solve and check equations and inequalities, and to model and analyze problems involving periodic motion, sound, light, and more.

d d x arcsin ⁡ ( x ) = 1 1 − x 2 \dfrac d x d ​ arcsin ( x ) = 1 − x 2 ​ 1 ​ start fraction, d, divided by, d, x, end fraction, \arcsin, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, square root of, 1, minus, x, squared, end square root, end fraction d d x arccos ⁡ ( x ) = − 1 1 − x 2 \dfrac d x d ​ arccos ( x ) = − 1 − x 2 ​ 1 ​ start fraction, d, divided by, d, x, end fraction, \arccos, left parenthesis, x, right parenthesis, equals, minus, start fraction, 1, divided by, square root of, 1, minus, x, squared, end square root, end fraction d/dx arccot(x) = - 1 / (1+x²) d/dx arcsec(x) = 1 / (x√(x²-1)) ; for 0≤x<π/2 and π≤x<3π/2 d/dx arcsec(x) = 1 / (|x|√(x²-1)) ; for 0≤x<π/2 and π/2

tan ⁡ ( L ) = opposite adjacent = 35 65 \tan(L) = \dfrac tan ( L ) = adjacent opposite ​ = 6 5 3 5 ​ tangent, left parenthesis, L, right parenthesis, equals, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, a, d, j, a, c, e, n, t, end text, end fraction, equals, start fraction, 35, divided by, 65, end fraction • Inverse sine ( sin ⁡ − 1 ) (\sin^) ( tan − 1 ) left parenthesis, tangent, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the tangent. Trigonometric functions input angles and output side ratios Inverse trigonometric functions input side ratios and output angles sin ⁡ ( θ ) = opposite hypotenuse \sin (\theta)=\dfrac \right)=\theta tan − 1 ( adjacent opposite ​ ) = θ tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, a, d, j, a, c, e, n, t, end text, end fraction, right parenthesis, equals, theta The expression sin ⁡ − 1 ( x ) \sin^ sin ( x ) 1 ​ start fraction, 1, divided by, sine, left parenthesis, x, right parenthesis, end fraction . In other words, the − 1 -1 − 1 minus, 1 is not an exponent. Instead, it simply means inverse function. A coordinate plane. The x-axis starts at zero and goes to ninety by tens. It is labeled degrees. The y-axis starts at zero and goes to two by two tenths. It is labeled a ratio. The graphed line is labeled sine of x, which is a nonlinear curve. The line f...

arcsin ⁡ ( x ) \arcsin(x) arcsin ( x ) \arcsin, left parenthesis, x, right parenthesis , or sin ⁡ − 1 ( x ) \sin^(x) sin − 1 ( x ) sine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis , is the inverse of sin ⁡ ( x ) \sin(x) sin ( x ) sine, left parenthesis, x, right parenthesis . arccos ⁡ ( x ) \arccos(x) arccos ( x ) \arccos, left parenthesis, x, right parenthesis , or cos ⁡ − 1 ( x ) \cos^(x) cos − 1 ( x ) cosine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis , is the inverse of cos ⁡ ( x ) \cos(x) cos ( x ) cosine, left parenthesis, x, right parenthesis . arctan ⁡ ( x ) \arctan(x) arctan ( x ) \arctan, left parenthesis, x, right parenthesis , or tan ⁡ − 1 ( x ) \tan^(x) tan − 1 ( x ) tangent, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis , is the inverse of tan ⁡ ( x ) \tan(x) tan ( x ) tangent, left parenthesis, x, right parenthesis . Radians Degrees − π 2 ≤ arcsin ⁡ ( θ ) ≤ π 2 -\dfrac − 2 π ​ < arctan ( θ ) < 2 π ​ minus, start fraction, pi, divided by, 2, end fraction, is less than, \arctan, left parenthesis, theta, right parenthesis, is less than, start fraction, pi, divided by, 2, end fraction − 9 0 ∘ < arctan ⁡ ( θ ) < 9 0 ∘ -90^\circ<\arctan(\theta)<90^\circ − 9 0 ∘ < arctan ( θ ) < 9 0 ∘ minus, 90, degrees, is less than, \arctan, left parenthesis, theta, right parenthesis, is less than, 90, degrees The trigonometric functions aren't really inverti...

d d x arcsin ⁡ ( x ) = 1 1 − x 2 \dfrac d x d ​ arcsin ( x ) = 1 − x 2 ​ 1 ​ start fraction, d, divided by, d, x, end fraction, \arcsin, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, square root of, 1, minus, x, squared, end square root, end fraction d d x arccos ⁡ ( x ) = − 1 1 − x 2 \dfrac d x d ​ arccos ( x ) = − 1 − x 2 ​ 1 ​ start fraction, d, divided by, d, x, end fraction, \arccos, left parenthesis, x, right parenthesis, equals, minus, start fraction, 1, divided by, square root of, 1, minus, x, squared, end square root, end fraction d/dx arccot(x) = - 1 / (1+x²) d/dx arcsec(x) = 1 / (x√(x²-1)) ; for 0≤x<π/2 and π≤x<3π/2 d/dx arcsec(x) = 1 / (|x|√(x²-1)) ; for 0≤x<π/2 and π/2