💯 Combining Trigonometric Functions and Inverse Trigonometric Functions

📌 Receive Comprehensive Mathematics Practice Papers Weekly for FREE 💯

Substitutions

When working with trigonometric functions, it is sometimes necessary to combine them in order to simplify expressions or solve equations. One common way to do this is to use trigonometric identities, which are equations that are true for all values of the variables involved.

For example, one commonly used identity is the Pythagorean identity:

sin^2(x) + cos^2(x) = 1

This identity can be used to simplify expressions involving sine and cosine functions. For example, if we have an expression like:

sin(x)cos(x)

We can use the identity to rewrite it as:

sin(x)cos(x) = sin(x)cos(x) * (sin^2(x) + cos^2(x))/(sin^2(x) + cos^2(x))

= sin(x)cos(x)sin^2(x)/ (sin^2(x) + cos^2(x)) + sin(x)cos(x)cos^2(x)/ (sin^2(x) + cos^2(x))

= sin^3(x)/ (sin^2(x) + cos^2(x)) + cos^3(x)/ (sin^2(x) + cos^2(x))

= (sin^3(x) + cos^3(x))/ (sin^2(x) + cos^2(x))

= (sin(x) + cos(x))(sin^2(x) - sin(x)cos(x) + cos^2(x))/ (sin^2(x) + cos^2(x))

= (sin(x) + cos(x))

So, we have simplified sin(x)cos(x) to sin(x) + cos(x) using the Pythagorean identity and some algebraic manipulation.

Inverse trigonometric functions, such as arccos, arcsin, and arctan, are used to find the angle whose trigonometric function is equal to a given value. For example, if we want to find the angle whose cosine is equal to 0.5, we can use the arccos function:

arccos(0.5) = cos^-1(0.5)

The value returned by the arccos function is the angle whose cosine is equal to 0.5. Similarly, the arcsin function is used to find the angle whose sine is equal to a given value, and the arctan function is used to find the angle whose tangent is equal to a given value.

In summary, combining trigonometric functions involves using trigonometric identities to simplify expressions involving sine, cosine, and other trigonometric functions. Inverse trigonometric functions are used to find the angle whose trigonometric function is equal to a given value.