Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. $\sin (a+b)=\sin a+\sin b$ b. The equation $\cos \theta =2$ has multiple real solutions. c. The equation $\sin \theta =\frac{1}{2}$ has exactly one solution. d. The function $\sin (\pi x/12)$ has a period of 12 e. Of the six basic trigonometric functions, only tangent and cotangent have a range of $(-\mathrm{\infty },\mathrm{\infty })$ f. $\frac{{\sin }^{-1}x}{{\cos }^{-1}x}={\tan }^{-1}x$ g. ${\cos }^{-1}(\cos (15\pi /16))=15\pi /16$ h. ${\sin }^{-1}x=1/\sin x$

Short answer

Question: Evaluate whether the following statement is true or false: ${\sin }^{-1}(x)=\frac{1}{\sin (x)}$. Answer: False

Step-by-step solution

a. Evaluating whether $\sin (a+b)=\sin a+\sin b$ is true

Using the angle addition formula, we have $\sin (a+b)=\sin a\cos b+\cos a\sin b$. This is not equal to $\sin a+\sin b$. Therefore, this statement is false. One example is by taking $a=\frac{\pi }{2}$ and $b=\frac{3\pi }{2}$, then $\sin (a+b)=\sin 2\pi =0$, but $\sin a+\sin b=1$.

b. Evaluating whether the equation $\cos \theta =2$ has multiple real solutions

Since the value of the cosine function lie in the range $[-1,1]$, it cannot equal 2 for any real number $\theta$. Thus, the statement is false.

c. Evaluating whether the equation $\sin \theta =\frac{1}{2}$ has exactly one solution

The sine function is periodic with a period of $2\pi$. This means that if $\sin \theta =\frac{1}{2}$, then $\sin (\theta +2\pi k)=\frac{1}{2}$ for any integer $k$. One solution is $\theta =\frac{\pi }{6}$ and another solution is $\theta =\frac{5\pi }{6}$, which are within one period of $\sin x$. Therefore, this statement is false.

d. Evaluating whether the function $\sin (\pi x/12)$ has a period of 12

The function has a period of $T$, if $\sin (\frac{\pi }{12}(x+T))=\sin (\frac{\pi }{12}x)$. This is true if $\frac{\pi }{12}(x+T)=\frac{\pi }{12}x+2\pi k$. In this case, the period is $T=24$. Therefore, this statement is false.

e. Evaluating whether only tangent and cotangent have a range of $(-\mathrm{\infty },\mathrm{\infty })$ among the six basic trigonometric functions

The six basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. The sine and cosine functions have a range of $[-1,1]$. The tangent and cotangent functions have a range of $(-\mathrm{\infty },\mathrm{\infty })$. The secant and cosecant functions, which are the reciprocals of cosine and sine respectively, also have a range of $(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$. Therefore, this statement is false.

f. Evaluating whether $\frac{{\sin }^{-1}x}{{\cos }^{-1}x}={\tan }^{-1}x$ is true

Let's use the fact that ${\tan }^{-1}x={\sin }^{-1}\frac{x}{\sqrt{1+x^2}}$. If we assume the statement to be true, then: $$\frac{{\sin }^{-1}x}{{\cos }^{-1}x}={\sin }^{-1}\frac{x}{\sqrt{1+x^2}}.$$ However, this identity is not true for all values of $x$. For example, consider $x=1$. In this case, we would have $\frac{\frac{\pi }{4}}{\frac{\pi }{2}-\frac{\pi }{4}}={\sin }^{-1}\frac{1}{\sqrt{2}}$, but this results in $\frac{1}{2}=\frac{\pi }{4}$, which is false. Therefore, this statement is false.

g. Evaluating whether ${\cos }^{-1}(\cos (\frac{15\pi }{16}))=\frac{15\pi }{16}$ is true.

The principal value range of ${\cos }^{-1}(x)$ is between $[0,\pi ]$ and $\frac{15\pi }{16}$ lies in that range. Therefore, ${\cos }^{-1}(x)$ is indeed the original angle itself, so the statement is true.

h. Evaluating whether ${\sin }^{-1}(x)=\frac{1}{\sin (x)}$ is true.

We can provide a simple counterexample to show this statement is false. Take $x=1$, then according to the statement, we get: ${\sin }^{-1}(1)=\frac{1}{\sin (1)}$. However, ${\sin }^{-1}(1)=\frac{\pi }{2}$ and $\frac{1}{\sin (1)}$ is not equal to $\frac{\pi }{2}$. Therefore, this statement is false.

Key concepts

Angle Addition Formulas

The angle addition formulas are incredibly useful identities in trigonometry. They help us find the sine, cosine, or tangent of the sum or difference of two angles. For the sine of the sum of two angles, the formula is $\sin (a+b)=\sin a\cos b+\cos a\sin b$. This formula clearly shows that $\sin (a+b)$ is not simply $\sin a+\sin b$. To illustrate this, consider when $a=\frac{\pi }{2}$ and $b=\frac{3\pi }{2}$, then $\sin (a+b)=\sin (2\pi )=0$, yet $\sin a+\sin b=1$. Such examples emphasize the importance of using the angle addition formulas properly.

Understanding these formulas not only eliminates common mistakes but also enhances the ability to tackle more complex trigonometric equations. Whenever you have a problem involving the sum or difference of angles, remember to use these identities to simplify your calculations.

Cosine Function Range

The cosine function, one of the fundamental trigonometric functions, has a specific range of $[-1,1]$ for real numbers. This means its maximum value is 1 and its minimum value is -1. It cannot exceed these bounds under normal circumstances. For instance, an equation like $\cos \theta =2$ doesn't have any solutions because 2 falls outside the cosine function's range.

This understanding is crucial as it guides which values of $\theta$ are possible when solving equations or constructing graphs involving the cosine function. Keeping the range of cosine in mind prevents errors in both calculations and logical reasoning involving cosine.

Sine Function Periodicity

The periodicity of the sine function is a fundamental concept in trigonometry. The sine function $\sin (x)$ has a period of $2\pi$, meaning that it repeats its values every $2\pi$ units. For example, if $\sin \theta =\frac{1}{2}$, then $\sin (\theta +2\pi k)=\frac{1}{2}$ for any integer $k$. This property implies that there are infinitely many solutions to $\sin \theta =\frac{1}{2}$, demonstrating that the initial conclusion of one solution is incorrect.

It's essential to understand the periodic nature of sine when solving problems or analyzing functions that involve sine. This knowledge not only aids in solving trigonometric equations but also in understanding real-world phenomena modeled by sinusoidal functions.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric value. Each inverse function has a specific range in which they provide unique results. For example, the range of ${\sin }^{-1}(x)$ is $[-\frac{\pi }{2},\frac{\pi }{2}]$, while the range of ${\cos }^{-1}(x)$ is $$0,\pi$$. These ranges reflect the principal value, ensuring that the inverse functions provide single-valued outputs.

In trigonometry, misunderstandings often involve these inverse functions. For instance, the incorrect assumption that ${\sin }^{-1}x=\frac{1}{\sin x}$ leads to confusion. It's imperative to appreciate the correct use of inverses to avoid logical errors. Recognizing how inverse trigonometric functions map angles back to their corresponding values can greatly enhance problem-solving abilities.