Section 7.4 Inverses of the Trigonometric Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives  Find values of the inverse trigonometric functions.  Simplify expressions such as sin (sin -1 x) and sin -1 (sin x).  Simplify expressions involving composition such as sin (cos –1 1/2) without using a calculator.  Simplify expressions such as sin arctan (a/b) by making a drawing and reading off appropriate ratios.

Inverse Sine Function The graphs of an equation and its inverse are reflections of each other across the line y = x. However, the inverse is not a function as it is drawn.

Inverse Sine Function We must restrict the domain of the inverse sine function. It is fairly standard to restrict it as shown here. The domain is [–1, 1] The range is [–π/2, π/2].

Inverse Cosine Function The graphs of an equation and its inverse are reflections of each other across the line y = x. However, the inverse is not a function as it is drawn.

Inverse Cosine Function We must restrict the domain of the inverse cosine function. It is fairly standard to restrict it as shown here. The domain is [–1, 1]. The range is [0, π].

The graphs of an equation and its inverse are reflections of each other across the line y = x. Inverse Tangent Function However, the inverse is not a function as it is drawn.

Inverse Tangent Function We must restrict the domain of the inverse tangent function. It is fairly standard to restrict it as shown here. The domain is (–∞, ∞). The range is (–π/2, π/2).

Inverse Trigonometric Functions FunctionDomainRange

Graphs of the Inverse Trigonometric Functions

Example Find each of the following function values. Find  such that sin  =. In the restricted range [–π/2, π/2], the only number with sine of is π/4. Solution:

Example (cont) Find  such that cos  = –1/2. In the restricted range [0, π], the only number with cosine of –1/2 is 2π/3.

Example (cont) Find  such that tan  = In the restricted range (–π/2, π/2), the only number with tangent of is –π/6.

Example Approximate the following function value in both radians and degrees. Round radian measure to four decimal places and degree measure to the nearest tenth of a degree. Solution: Press the following keys (radian mode): Readout: Rounded: Rounded: 105.6º Change to degree mode and press the same keys: Readout:

Composition of Trigonometric Functions for all x in the domain of sin –1 for all x in the domain of cos –1 for all x in the domain of tan –1

Example Simplify each of the following. Solution: a) Since is in the domain, [–1, 1], it follows that b) Since 1.8 is not in the domain, [–1, 1], we cannot evaluate the expression. There is no number with sine of 1.8. So, sin (sin –1 1.8) does not exist.

Special Cases for all x in the range of sin –1 for all x in the range of cos –1 for all x in the range of tan –1

Example Simplify each of the following. Solution: a) Since π/6 is in the range, (–π/2, π/2), it follows that b) Since 3π/4 is not in the range, [–π/2, π/2], we cannot apply sin –1 (sin x) = x.

Example Find Solution: cot –1 is defined in (0, π), so consider quadrants I and II. Draw right triangles with legs x and 2, so cot  = x/2.