promath Note that sin(cos-1 x) or sin(arc cosx) are algebraic functions, not trigonometric functions. The value of sin(cos-1 x) or sin(arc cosx) is equal to $\sqrt{1-x^2}$. In this post, we will simplify sin(cos^{-1} x). The formula of sin(cos inverse x) is given below: sin(cos-1 x) = root(1-x2). Simplify sin(cos-1(x)) We all know that $\sin(\cos^{-1}(x))= \sin(\text{arc} \cos … Read more
The value of cos(sin^-1(x)) or cos(arc sin x) is equal to $\sqrt{1-x^2}$. In this post, we will simplify the expression $\cos(\sin^{-1}(x))$. Simplify cos(sin-1(x)) We note that both $\cos(\sin^{-1}(x))$ and $\cos(arc \sin x)$ have the same value. Thus to find out the value of $\cos(\sin^{-1}(x))$ or $\cos(arc \sin x)$, let us put $\theta=\sin^{{-1}} x$ $\cdots (\star)$ … Read more
In this post, we will find the integral of root(a2-x2). The integration of square root of a2-x2 is given as follows ∫ √(a2-x2) dx = x/2 ⋅ √(a2-x2) + a2/2 sin-1(x/a) + C, where C is an integration constant. Integration of $\sqrt{a^2-x^2}$ Let $I = \int \sqrt{a^2-x^2} dx$ $\cdots (I)$ We will find the integration … Read more
The derivative of 2x is equal to 2xln2 where ln 2 is the natural logarithm of 2, that is, ln 2 = loge2. In this post, we will find the derivative of 2x by the first principle of derivatives. Derivative of 2x from First Principle We know that the derivative of a function f(x) by … Read more
In this blog post, we will learn how to simplify the trigonometric expressions cos4 x -sin4 x and cos4x+sin4x. We will apply the following two trigonometric identities: cos4x-sin4x Formula To simplify the expression cos4 x -sin4 x, we first apply the formula a2-b2 =(a-b)(a+b) with $a=\cos^2x$ and $b=\sin^2x$. Therefore, $\cos^4x-\sin^4x$ $=(\cos^2x)^2 – (\sin^2 x)^2$ $=(\cos^2x-\sin^2x)(\cos^2x+\sin^2x)$ … Read more
The derivative of xex is equal to (1+x)ex. In this post, we will find the derivative of xex by the first principle and by the product rule of derivatives. The first principle of derivatives says that the derivative of a function f(x) is given by the following limit formula: $\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ … Read more
The integration of 1/(1+x2) is equal to tan2 x. In this post, we will see how to integrate 1/(1+x^2). Integration of $\dfrac{1}{1+x^2}$ Question: What is the integration of $\dfrac{1}{1+x^2}$? That is, Find $\int \dfrac{1}{1+x^2} dx$ Answer: The integration of $\dfrac{1}{1+x^2}$ is $\tan^2 x$. Explanation: Let us substitute $x=\tan t$ $\cdots (\star)$ Differentiating with respect to … Read more
Yes, $\dfrac{1}{3}$ is a rational number. In this post, we will learn how to prove $\dfrac{1}{3}$ is a rational number. At first, we first recall the definition of a rational number. Definition of Rational Number A number $x$ is called a rational number if it can be represented as $\dfrac{p}{q}$ where $p$ and $q$ are … Read more
The derivative of 10x is equal to 10x ln 10. Here ln denotes the logarithm with base e, called the natural logarithm. In this post, we will learn to compute the derivative of 10 raised to x. Derivative of 10x Question: What is the derivative of 10x? Answer: The derivative of 10x is 10x ln … Read more
The value of the expression 1-sin^2(x) is equal to cos^2(x). In this post, we will find the formula of 1-sin^2x. 1-sin2x Formula To simplify the expression 1 – sin2x, we will follow the below steps: Step 1: Let us apply the following Pythagorean trigonometric identity: 1 = sin2x + cos2x Step 2: Now, we substitute the above … Read more