The Main Idea

Inverse trigonometric functions “undo” sine, cosine, and tangent. But since sine, cosine, and tangent are periodic, they are not one-to-one over their entire domains. To make their inverses valid functions, we must restrict the domain of each trig function to an interval where it is one-to-one and covers all possible outputs. These restricted intervals are chosen so the inverse functions give a single, consistent answer.

Quick Tips: Domain Restrictions for Inverse Trig

1. Inverse Sine ([latex]y = \sin^{-1}(x)[/latex])

   • Domain: [latex]-1 \leq x \leq 1[/latex]

   • Range: [latex]-\dfrac{\pi}{2} \leq y \leq \dfrac{\pi}{2}[/latex]

   • Chosen because sine is one-to-one in Quadrants I and IV.

2. Inverse Cosine ([latex]y = \cos^{-1}(x)[/latex])

   • Domain: [latex]-1 \leq x \leq 1[/latex]

   • Range: [latex]0 \leq y \leq \pi[/latex]

   • Chosen because cosine is one-to-one in Quadrants I and II.

3. Inverse Tangent ([latex]y = \tan^{-1}(x)[/latex])

   • Domain: all real numbers [latex](-\infty, \infty)[/latex]

   • Range: [latex]-\dfrac{\pi}{2} < y < \dfrac{\pi}{2}[/latex]

   • Chosen because tangent is one-to-one in Quadrants I and IV.

The Main Idea

Inverse trig functions return a principal angle whose trig value matches the input. Because sine, cosine, and tangent are periodic, their inverses must be restricted to certain ranges so they give a single, consistent answer. To find exact values, use the special triangles ([latex]30^\circ[/latex]–[latex]60^\circ[/latex]–[latex]90^\circ[/latex] and [latex]45^\circ[/latex]–[latex]45^\circ[/latex]–[latex]90^\circ[/latex]) or the unit circle. For composite expressions like [latex]\sin(\cos^{-1} x)[/latex] or [latex]\tan(\sin^{-1} x)[/latex], build a right triangle from the inner inverse, then use it to evaluate the requested function — always checking the sign based on the inverse’s range.

Quick Tips: Exact Value with Inverse Sine, Cosine, and Tangent

1. Know the Principal Ranges:

   • [latex]\sin^{-1}(x) \in \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right][/latex]

   • [latex]\cos^{-1}(x) \in \left[0, \pi\right][/latex]

   • [latex]\tan^{-1}(x) \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)[/latex]

2. Special Angles:

   • [latex]\sin^{-1}\left(\dfrac{1}{2}\right)=\dfrac{\pi}{6}[/latex]

   • [latex]\cos^{-1}\left(\dfrac{1}{2}\right)=\dfrac{\pi}{3}[/latex]

   • [latex]\tan^{-1}(1)=\dfrac{\pi}{4}[/latex]

The Main Idea

Inverse trigonometric functions allow us to find the angle when we know the value of sine, cosine, or tangent. Since most values do not correspond to “special angles,” we rely on a calculator for decimal approximations. The calculator gives the principal value of the inverse function, which comes from the restricted range of each inverse. To get correct results, the most important step is setting the calculator to the right mode: degrees if the answer should be in degrees, or radians if the answer should be in radians.

Quick Tips: Using a Calculator for Inverse Trig

1. Check the Mode First

   • Degree mode if you want an angle in degrees.

   • Radian mode if you want an angle in radians.

2. Use the Inverse Keys

   • Most calculators label them as [latex]\sin^{-1}[/latex], [latex]\cos^{-1}[/latex], [latex]\tan^{-1}[/latex].

   • Enter the number first or press the function button first (depends on calculator).

3. Principal Values Only

   • [latex]\sin^{-1}(x)[/latex] returns an angle in [latex]\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right][/latex].

   • [latex]\cos^{-1}(x)[/latex] returns an angle in [latex]\left[0, \pi\right][/latex].

   • [latex]\tan^{-1}(x)[/latex] returns an angle in [latex]\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)[/latex].

The Main Idea

When solving right triangles, we often know two sides and want to find an angle. Inverse trigonometric functions let us “work backward” from side ratios to angles. For example, if we know the opposite and adjacent sides, the ratio [latex]\dfrac{\text{opp}}{\text{adj}}[/latex] gives tangent, so we use [latex]\tan^{-1}[/latex] to find the angle. This process is especially useful in applications like surveying, construction, and navigation where angles must be determined from measurements of side lengths.

Quick Tips: Solving Right Triangles with inverse Trig

1. Choose the Right Function

   • Use [latex]\sin^{-1}[/latex] when you know opposite and hypotenuse.

   • Use [latex]\cos^{-1}[/latex] when you know adjacent and hypotenuse.

   • Use [latex]\tan^{-1}[/latex] when you know opposite and adjacent.

2. Set Up the Ratio

3. Apply the Inverse Function

4. Find the Other Angle

   • In a right triangle, the acute angles sum to [latex]90^\circ[/latex] (or [latex]\dfrac{\pi}{2}[/latex]).

   • If one angle is found with inverse trig, subtract from [latex]90^\circ[/latex] (or [latex]\dfrac{\pi}{2}[/latex]) to get the other.

The Main Idea

Composite expressions like [latex]\sin(\cos^{-1} x)[/latex] or [latex]\tan(\sin^{-1} x)[/latex] ask us to evaluate one trig function of an inverse trig function. These look complicated, but the strategy is simple: treat the inner inverse as an angle, build a right triangle that matches it, and then use the triangle to find the requested function value. The restricted ranges of inverse trig functions guarantee the angle is in a specific quadrant, which tells us the correct sign.

Quick Tips: Composite Functions with Inverse Trig

1. Think of the Inner Function as an Angle

   • Example: Let [latex]\theta = \cos^{-1}(x)[/latex]. Then [latex]\cos \theta = x[/latex].

2. Build a Right Triangle

   • Use the ratio given by sine, cosine, or tangent to assign sides of a right triangle.

   • Fill in the missing side with the Pythagorean Theorem.

3. Evaluate the Outer Function

   • Use the triangle to find the required trig value.

   • Example: [latex]\sin(\cos^{-1} x) = \sqrt{1-x^2}[/latex].

4. Check the Quadrant

   • [latex]\sin^{-1}(x)[/latex] gives an angle in Quadrants I or IV.

   • [latex]\cos^{-1}(x)[/latex] gives an angle in Quadrants I or II.

   • [latex]\tan^{-1}(x)[/latex] gives an angle in Quadrants I or IV.

   • This determines whether the answer is positive or negative.

5. Common Results to Remember

   • [latex]\sin(\cos^{-1} x) = \sqrt{1-x^2}[/latex]

   • [latex]\cos(\sin^{-1} x) = \sqrt{1-x^2}[/latex]

   • [latex]\tan(\sin^{-1} x) = \dfrac{x}{\sqrt{1-x^2}}[/latex]

   • [latex]\tan(\cos^{-1} x) = \dfrac{\sqrt{1-x^2}}{x}[/latex]

6. Worked Examples

   • [latex]\sin(\cos^{-1}(\dfrac{3}{5})) = \dfrac{4}{5}[/latex]

   • [latex]\tan(\sin^{-1}(\dfrac{5}{13})) = \dfrac{5}{12}[/latex]

   • [latex]\cos(\tan^{-1}(-\dfrac{\sqrt{3}}{3})) = \dfrac{\sqrt{3}}{2}[/latex]