Write the trigonometric expression cos(sin^(-1) x - cos^(-1) y) as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function.

Question
Trigonometric Functions
asked 2020-11-26
Write the trigonometric expression \(\displaystyle{\cos{{\left({{\sin}^{{-{1}}}{x}}-{{\cos}^{{-{1}}}{y}}\right)}}}\) as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function.

Answers (1)

2020-11-27
The difference between the two angles based on the cosine terms is given by,
\(\displaystyle{\cos{{\left(\alpha-\beta\right)}}}={\cos{\alpha}}{\cos{\beta}}+{\sin{\alpha}}{\sin{\beta}}\)
Substituting the values of alpha and beta in the formula,
\(\displaystyle{\cos{{\left(\alpha-\beta\right)}}}={\cos{\alpha}}{\cos{\beta}}+{\sin{\alpha}}{\sin{\beta}}\)
\(\displaystyle{\cos{{\left({{\sin}^{{-{1}}}{x}}-{{\cos}^{{-{1}}}{y}}\right)}}}={\cos{{\left({{\sin}^{{-{{1}}}}{x}}\right)}}}{\cos{{\left({{\cos}^{{-{1}}}{y}}\right)}}}+{\sin{{\left({{\sin}^{{-{1}}}{x}}\right)}}}{\sin{{\left({{\cos}^{{-{1}}}{y}}\right)}}}\)
Hence,
\(\displaystyle{\sin{{\left({{\sin}^{{-{1}}}{x}}\right)}}}={x}{\quad\text{and}\quad}{\cos{{\left({{\cos}^{{-{1}}}{y}}\right)}}}={y}\)
On simplifying,
\(\displaystyle{\cos{{\left({{\sin}^{{-{1}}}{x}}\right)}}}{\cos{{\left({{\cos}^{{-{1}}}{y}}\right)}}}{\sin{{\left({{\sin}^{{-{1}}}{x}}\right)}}}{\cos{{\left({{\cos}^{{-{1}}}{y}}\right)}}}\)
On applying the Pythagorean identities as shown below,
\(\displaystyle{\cos{{\left({{\sin}^{{-{1}}}{x}}\right)}}}{\quad\text{and}\quad}{\sin{{\left({{\cos}^{{-{1}}}{y}}\right)}}}\)
\(\displaystyle{\cos{{\left({{\sin}^{{-{1}}}{x}}\right)}}}{y}+{x}{\sin{{\left({{\cos}^{{-{1}}}{y}}\right)}}}=\sqrt{{{1}-{{\sin}^{{2}}{\left({{\sin}^{{-{1}}}{x}}\right)}}}}{y}+{x}\sqrt{{{1}-{{\cos}^{{2}}{\left({{\cos}^{{-{1}}}{y}}\right)}}}}\)
On simplification the equation obtained is,
\(\displaystyle\sqrt{{{1}-{{\sin}^{{2}}{\left({{\sin}^{{-{1}}}{x}}\right)}}}}{y}+{x}\sqrt{{{1}-{{\cos}^{{2}}{\left({{\cos}^{{-{1}}}{y}}\right)}}}}=\sqrt{{{1}-{x}^{{2}}}}{y}+{x}\sqrt{{{1}-{y}^{{2}}}}\)
\(\displaystyle={y}\sqrt{{{1}-{x}^{{2}}}}+{x}\sqrt{{{1}-{y}^{{2}}}}\)
Hence,the simplified expression is \(\displaystyle{y}\sqrt{{{1}-{x}^{{2}}}}+{x}\sqrt{{{1}-{y}^{{2}}}}\)
0

Relevant Questions

asked 2021-02-25
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function.
Given:
\(\displaystyle \tan{{\left({{\cos}^{ -{{1}}}{5}}{x}\right)}}=?\)
asked 2021-03-06
Given the values for sin t and cos t, use reciprocal and quotient identities to find the values of the other trigonometric functions of t.
\(\displaystyle{\sin{{t}}}=\frac{{3}}{{4}}{\quad\text{and}\quad}{\cos{{t}}}=\frac{\sqrt{{7}}}{{4}}\)
asked 2021-01-23
The two trigonometric functions defined for all real numbers are the_________ function and the_______ function. The domain of each of these functions is________ .
asked 2021-02-13
The two trigonometric functions defined for all real numbers are the_________ function and the_______ function. The domain of each of these functions is________ .
asked 2021-02-09
Given the following information about one trigonometric function, evaluate the other five functions.
\(\displaystyle{\cos{{u}}}=\frac{{5}}{{13}}\) , where \(\displaystyle{0}\le{u}\le\frac{\pi}{{2}}.\)
asked 2021-03-06
Sketch a right triangle corresponding to the trigonometric function of the acute angle theta. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of theta. \(\displaystyle{\cos{\theta}}=\frac{{21}}{{5}}\)
asked 2021-02-25
To further justify the Cofunction Theorem, use your calculator to find a value for the given pair of trigonometric functions. In each case, the trigonometric functions are cofunctions of one another, and the angles are complementary angles. Round your answers to four places past the decimal point.
\(\displaystyle{{\sec{{6.7}}}^{\circ},}{\cos{{e}}}{c}{83.3}^{\circ}\)
asked 2021-01-05
Find derivative of trigonometric function \(\displaystyle{y}=\frac{{{3}{\left({1}-{\sin{{x}}}\right)}}}{{{2}{\cos{{x}}}}}\)
asked 2021-02-08
Find derivative of trigonometric function \(\displaystyle{y}=\frac{{{3}{\left({1}-{\sin{{x}}}\right)}}}{{{2}{\cos{{x}}}}}\)
asked 2020-11-05
How are the inverse trigonometric functions defined?
...