Show that for any numbers a and b, the sine inequality |sin b - sin a|<=| b         -a | is true.

Q: Prove that the equation sinx = 0.3 has at least one solution in the interval (o, ~).

A: Given:sinx=0.3

Q: Please solve a question on hyperbolic functions.

A: Let us first find x + iy and its complex conjugate x – iy. Complex conjugate can be found by…

Q: 2.) Evaluate using wallis formula 1-x² dx and let x= sine

A: Given ∫01x41-x2dx By using 1-sin2(x)=cos2x And Wallis formula…

Q: (-n *) Prove sin? x – sinx cosx – 2 cos? x =0 in п, 2 has five root's Hint. Intermediate value…

A:

Q: 5. Given that 2 sin 20 = cos 20, a) show that tan 20 = 0.5. b) hence find the values of 0, to one…

A: given that 2sin2θ=cos2θ to prove that tan2θ=0.5 proceed as follows  2sin2θcos2θ=1tan2θ=12tan2θ=0.5

Q: Prove that: (1 – sin? x)(1 + tan? x) = 1

A:

Q: Prove the identity. +x + sin 6. sin = cos x

A: Given, We have to prove the identity sinπ6+x+sinπ6-x=cosx

Q: (a) Prove that 1- cos 2.x NT = tan x, sin 2.x (b) Hence solve, for 0 < 0 < 2n, 1- cos 20 3sec? 0 – 7…

A: Solution(a): Prove that 1-cos2xsin2x=tanx LHS:1-cos2xsin2x=1-1-2sin2x2sinxcosx…

Q: Find all values of x in the interval [0, 2?] that satisfy the equation. 6 cos(x) + 3 sin(2x) = 0

A:

Q: Prove the following inequalities (ii) -10|< sin 0 < |0| and -|0| < 1– cos 0 < |0|

A:

Q: Use the mean value theorem to prove the inequality for all a and b: Isina – sinb| < |a - b|.

A:

Q: Prove the identity sin(-0)- cos (+0)

A:

Q: Find all exact solutions to 10-sin(x)– 4= 18 sin(x) in the interval [0,2 1)

A:

Q: 1. Use the Mean Value theorem to show that for any two real numbers a and b, we have | sin b - sin a…

A:

Q: Find the exact value of sin 5π/12 using the fact that 5π/12 = π/6 + π/4.

A:

Q: Prove that the equation 4 sin 4 has a solution.

A: Given,

Q: Prove the following inequalities -10|< sin 0 < |0| and -10| < 1– cos 0 < |0|

A: Given the inequalities         -θ≤sin θ≤θ and -θ≤1-cos θ≤θ. Prove it.

Q: Prove that the inequality | cos(x)| > 1 – sin? (x) holds true for all aE R.

A:

Q: B) Prove that: 1-tan? 0 cos? 0 – sin? 0 1+tan? 0

A:

Q: Find sin2x, cos2x, and tan 2x if cosx - and x terminates in guadrant III. 13 sin 2x cos 2x tan 2x

A:

Q: Find sin 5, given that sin x = 2 i and 0<x<5.

A: Here  sin x=14⇒cos x=1-sin2x              ⇒cos x=1-116              ⇒cos x=154 Now, sin2x2=1- cos x2…

Q: Solve the following inequality on the interval [0, 7]. 2 sin (0) + 3 cos(20) < 0

A: topic - inequality

Q: Prove that the inequality                          | cos(x) | ≥ 1 − sin2(x) holds true for all x ∈…

A: we have to prove that the inequality  cos(x)≥1-sin2x holds true for all x ∈R from given , we have…

Q: It is known that sin 36°=sin x and xe[0°,360°] . Write down all the possible value(s) of x

A:

Q: Hence prove that sin?a + cos²ɑ = 1 3.2

A: Since you have posted more than one question, we solve the first one for you. For the other…

Q: 1 Use the inequality sin xs x, which holds for x20, to find an upper bound for the value of sin x…

A:

Q: . Use the Mean Value Theorem to prove the inequality. sin a sin b| < Ja – b Va and b -

A:

Q: V3 Prove that sin 10° sin50° sin 60° sin 70° %3D 16

A: Given, sin10.sin50.sin60.sin70 =sin 60.sin10.sin50.sin70 = [(√3)/2]sin10sin50sin70 =…

Q: Find a particular solution to 3 " sin (3t). 2 -3 5

A:

Q: tan 5, given sin 0: 3 270° <0 < 360° 5

A:

Q: B) Prove that: cos? 0 – sin? 0 1-tan? 0 =- 1+tan? 0

A: B)cos2θ-sin2θ=1-tan2θ1+tan2θRHS1-tan2θ1+tan2θ

Q: show that there is a root of sin(x2)=e2-2.

A: To show that there is a root of sin(x2)=e2-2.

Q: Show that given that z = sin-12 дхду дудх

A:

Q: Find at least one solution to the following equation: sin(x^2 − 1) / (1 − sin(x^2 − 1) )= sin(x) +…

A:

Q: Use the inequality sin xsx, which holds for x20, to find an upper bound for the value of sin x dx.…

A:

Q: Solve the following inequality on the interval 0, 7. 2 sin (A) + 3 cos(20) <0

A:

Q: Prove the following identity. (sin x − cos x)2 = 1 − sin 2x   solve from the right

A: Expand lhs for proven

Q: Solve for all values of x over the interval [0,27]: 2sin(x) = sin(2x)

A: 2 sin x = sin (2x)             = 2 sin x cos x Hence 2 sin x - 2 sin x cos x = 0  2 sin x ( 1 - cos…

Q: Using De Moivre's Theorem, show that GUTM n 0 – 20 sin³ 0 + 5 sin 0. UTM sin 50 = 16 sin 3 | Hence,…

A:

Q: Prove that there is a real number r so that sin(r) = x – 1.

A: We have, sinx=x-1 Let us consider a function fx such that, fx=x-1-sinx

Q: Prove that the inequality | cos(x)| 21- sin?(x) holds true for all æ e R.

A:

Q: What mathematical result is applied to prove that sin^2(e)+cos^2(0)=1?

A:

Q: Prove that the following identity is true.   sin4 t - cos4 t / sin2 t cos2 t = sec2 t - csc2 t

A: Given that sin4t-cos4tsin2tcos2t=sec2t-cosec2t

Q: Let sin A = 3/5 with A   sin 2 A

A: Given:sin A=35Formula:sin 2A=2 sinA cosAsin2A=1-cos2A

Q: Find all solutions on the interval [0,27] for sin(20) = 2.

A: topic- inverse trigonometric functions

Q: Find all solutions to 2 sin(0) = 2 on the interval 0 <0 < 2n.

A:

Q: Prove that; (a) cos4θ- sin4θ = cos 2θ  (b) Sin (π/4 + A) + Sin (π/4 - A) ≡ √2 cos A

A:

Q: Prove: sin? xcos?x = - cos(4x). 8.

A: We will use various trigonometric formulas to prove the given identity.

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.