Question

Right-triangle relationships Use a right triangle to simplify the given expressions. Assume $x>0.$ $$\sin ({\sec }^{-1}\left(\frac{\sqrt{x^2+16}}{4}\right))$$

Short answer

Question: Simplify the expression $\sin ({\sec }^{-1}(\frac{\sqrt{x^2+16}}{4}))$. Answer: $$\frac{x}{\sqrt{x^2+16}}.$$

Step-by-step solution

Determine the right triangle based on the inverse secant function

Since we are asked to find $\sin ({\sec }^{-1}(\frac{\sqrt{x^2+16}}{4}))$, first we need to define the angle $\theta$ such that: $\sec (\theta )=\frac{\sqrt{x^2+16}}{4}$. Recall that $\sec (\theta )=\frac{1}{\cos (\theta )}$, so we have $\cos (\theta )=\frac{4}{\sqrt{x^2+16}}$. Now, we will use this information to find a right triangle with angle $\theta$. We know that $\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}$. Letting the adjacent side be 4 and the hypotenuse be $\sqrt{x^2+16}$, we can form the right triangle.

Determine the opposite side of the right triangle

We have the adjacent side (4) and the hypotenuse ($\sqrt{x^2+16}$) of the right triangle. We will now find the length of the opposite side using the Pythagorean theorem: $a^2+b^2=c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse of the right triangle. In our case, we have: $4^2+b^2=(\sqrt{x^2+16}{)}^2$. Simplifying, we have: $16+b^2=x^2+16$. Thus, we find that the opposite side (b) is equal to: $b^2=x^2$ $b=x$.

Calculate the sine of the angle θ

Now that we have the three sides of the right triangle, we can find the sine of the angle θ. We know that $\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$. In our triangle, the opposite side is $x$ and the hypotenuse is $\sqrt{x^2+16}$. So the sine of the angle θ is: $\sin (\theta )=\frac{x}{\sqrt{x^2+16}}$.

Substitute the value of $\sin (\theta )$ into the given expression

Finally, we substitute the value of $\sin (\theta )$ back into the given expression: $\sin ({\sec }^{-1}(\frac{\sqrt{x^2+16}}{4}))=\sin (\theta )=\frac{x}{\sqrt{x^2+16}}$. So, the simplified expression is: $$\frac{x}{\sqrt{x^2+16}}.$$

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions provide a method to determine the angle that corresponds to a given trigonometric ratio. In many mathematical and physical problems, we often know the ratios of sides in a right triangle but need to find the measure of the angle. For example, if we know the ratio of the hypotenuse to the adjacent side (the secant of an angle), we can use the arcsecant (or inverse secant, denoted as ${\sec }^{-1}$) to find the angle itself.

In the given exercise, the use of ${\sec }^{-1}$ helped us find the angle, $\theta$, for which the secant is $\frac{\sqrt{x^2+16}}{4}$. Understanding inverse trigonometric functions is crucial in simplifying such expressions, as they act as a 'bridge' from a ratio of sides back to angles.

Right Triangle Relationships

The relationships within a right triangle are the foundation for trigonometry. A right triangle has one angle that is exactly 90 degrees, and two other angles that are acute. The sides of a right triangle are named the 'hypotenuse' (the side opposite the right angle and also the longest side), the 'opposite' (the side opposite the angle we are focused on), and the 'adjacent' (the side next to the angle we are focused on but not the hypotenuse).

Trigonometric functions like sine, cosine, and tangent relate the angles to the lengths of the sides. For instance, cosine is defined as the adjacent side divided by the hypotenuse, and sine is the opposite divided by the hypotenuse. These relationships are used to solve for unknown sides or angles in right triangles, as seen in the exercise where they facilitated finding the length of the side opposite to angle $\theta$.

Pythagorean Theorem

The Pythagorean theorem is one of the most famous principles in geometry. It states that for any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). The theorem can be written as $a^2+b^2=c^2$, where $c$ is the hypotenuse, and $a$ and $b$ are the other two sides.

In the context of the exercise, this theorem enabled us to find the unknown side of the right triangle by expressing the lengths of the sides as a relationship: when we knew the lengths of the hypotenuse and one leg (the adjacent side), we could solve for the length of the other leg (the opposite side) using this principle. The Pythagorean theorem is a powerful tool for working with right triangles, and it's essential for all students to master.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that compare the lengths of sides of a right triangle relative to one of its acute angles. As mentioned, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, $\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$, while the cosine is the ratio of the length of the adjacent side to the hypotenuse, $\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}$.

These concepts proved essential in the exercise to reverse engineer the problem: given the secant (which is the reciprocal of cosine), we determined the cosine, then found the opposite side with the Pythagorean theorem, and then used the sine function to find the ratio of the opposite to the hypotenuse. Mastering the relationships between sine, cosine, and the sides of a triangle is incredibly helpful in trigonometry.