Podcast
Questions and Answers
If $x$, $y$, and $z$ are in Arithmetic Progression, and $\tan x$, $\tan y$, and $\tan z$ are also in Arithmetic Progression, where $x, z > 0$ and $xz < 1$, $y < \frac{1}{2}$, $z < 1$, which of the following relationships must be true?
If $x$, $y$, and $z$ are in Arithmetic Progression, and $\tan x$, $\tan y$, and $\tan z$ are also in Arithmetic Progression, where $x, z > 0$ and $xz < 1$, $y < \frac{1}{2}$, $z < 1$, which of the following relationships must be true?
- $6x = 4y = 3z$
- $2x = 3y = 6z$
- $6x = 3y = 2z$
- $x = y = z$ (correct)
Given that $\tan(x+2) + \tan(x-2) - \tan(\frac{1}{2}) = 0$, what is one possible value of $x$?
Given that $\tan(x+2) + \tan(x-2) - \tan(\frac{1}{2}) = 0$, what is one possible value of $x$?
- 1 (correct)
- 2
- -1
- $\frac{1}{2}$
If an angle $\theta$ in the interval $[0, 2\pi]$ satisfies both $\cot \theta = \sqrt{3}$ and $\sqrt{3} \sec \theta + 2 = 0$, then what is the value of $\theta$?
If an angle $\theta$ in the interval $[0, 2\pi]$ satisfies both $\cot \theta = \sqrt{3}$ and $\sqrt{3} \sec \theta + 2 = 0$, then what is the value of $\theta$?
- $\frac{11\pi}{6}$
- $\frac{5\pi}{6}$
- $\frac{7\pi}{6}$ (correct)
- $\frac{\pi}{6}$
In a triangle, the angles are in the ratio 5:1:6. What is the ratio of the smallest side to the greatest side?
In a triangle, the angles are in the ratio 5:1:6. What is the ratio of the smallest side to the greatest side?
If the sides $a, b, c$ of a triangle are in Arithmetic Progression, what is the value of $a \cos^2(\frac{A}{2}) + c \cos^2(\frac{C}{2})$?
If the sides $a, b, c$ of a triangle are in Arithmetic Progression, what is the value of $a \cos^2(\frac{A}{2}) + c \cos^2(\frac{C}{2})$?
If $\cos x = a$ where $0 < x < 1$, and $\sin^{-1}(x\sqrt{1-x^2}) + \sec^{-1}(\frac{1}{2x^2-1}) = \frac{\pi}{3}$, determine the expression for $a$.
If $\cos x = a$ where $0 < x < 1$, and $\sin^{-1}(x\sqrt{1-x^2}) + \sec^{-1}(\frac{1}{2x^2-1}) = \frac{\pi}{3}$, determine the expression for $a$.
If $0 < x < 1$, what is the simplified form of $\sqrt{1 + x^2} \cdot [{\cos(\cot^{-1}x) + \sin(\cot^{-1}x)} ^{-1}]^{\frac{1}{2}}$?
If $0 < x < 1$, what is the simplified form of $\sqrt{1 + x^2} \cdot [{\cos(\cot^{-1}x) + \sin(\cot^{-1}x)} ^{-1}]^{\frac{1}{2}}$?
In a triangle ABC with usual notations, what expression is equivalent to $2ac \sin^2(\frac{A - B + C}{2})$?
In a triangle ABC with usual notations, what expression is equivalent to $2ac \sin^2(\frac{A - B + C}{2})$?
Determine the number of values of $x$ in the interval $(0, 5\pi)$ satisfying the equation $3\sin^2x - 7\sin x + 2 = 0$.
Determine the number of values of $x$ in the interval $(0, 5\pi)$ satisfying the equation $3\sin^2x - 7\sin x + 2 = 0$.
If $\tan^{-1}(\frac{1}{4}) + \tan^{-1}(\frac{1}{9}) = \frac{1}{2}\cos^{-1}x$, then what is the value of $x$?
If $\tan^{-1}(\frac{1}{4}) + \tan^{-1}(\frac{1}{9}) = \frac{1}{2}\cos^{-1}x$, then what is the value of $x$?
Given $\cos^{-1}(\frac{12}{13}) = \sin^{-1}P$, what is the value of $P$?
Given $\cos^{-1}(\frac{12}{13}) = \sin^{-1}P$, what is the value of $P$?
In a triangle $ABC$, if $I(AB) = \sqrt{23}$ units, $I(BC) = 3$ units, and $I(CA) = 4$ units, then find the value of $\frac{\cot A + \cot C}{\cot B}$.
In a triangle $ABC$, if $I(AB) = \sqrt{23}$ units, $I(BC) = 3$ units, and $I(CA) = 4$ units, then find the value of $\frac{\cot A + \cot C}{\cot B}$.
Flashcards
Arithmetic Progression of Tangents
Arithmetic Progression of Tangents
If x, y, z are in Arithmetic Progression (AP) and tan x, tan y, tan z are also in AP, with x, z > 0 and xz < 1, and y < 1, then 6x = 3y= 2z
Angle satisfying trigonometric equations
Angle satisfying trigonometric equations
Angle (\theta) in [0, 2(\pi)] that satisfies both cot (\theta) = (\sqrt{3}) and (\sqrt{3}) sec (\theta) + 2 = 0 is 7(\pi)/6.
Ratio of triangle sides
Ratio of triangle sides
If the angles of a triangle are in the ratio 5:1:6, the ratio of the smallest side to the greatest side is (\sqrt{3}) - 1 : 2(\sqrt{2}).
Triangle sides in AP
Triangle sides in AP
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a in Trigonometric Equation
a in Trigonometric Equation
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Trigonometric simplification
Trigonometric simplification
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Triangle Identity
Triangle Identity
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Study Notes
Trigonometric Functions
- If x, y, z are in Arithmetic Progression (AP) and tan x, tan y, tan z are also in AP, where x, z > 0 and xz < 1, y < 1, then 6x = 3y = 2z.
- If tan⁻¹(x + 2) + tan⁻¹(x - 2) = tan⁻¹(1/2), then x = 1/2.
- If angle θ in [0, 2π] satisfies cot θ = √3 and √3 sec θ + 2 = 0, then θ = 7π/6.
- If the angles of a triangle are in the ratio 5:1:6, the ratio of the smallest side to the greatest side is √3 - 1 : 2√2.
- If the sides of a triangle a, b, c are in A.P., then with usual notations, a cos²(A/2) + c cos²(C/2) = 3b/2.
- If cos⁻¹x = a (0 < x < 1) and sin⁻¹(√(1 - x²)) + sec⁻¹(1/(2x² - 1)) = π/2, then a = 1/√2.
- If 0 < x < 1, then √1 + x²[{x cos(cot⁻¹x) + sin(cot⁻¹x)}²]⁻¹/² is equal to x.
- In a triangle ABC, with usual notations, 2ac sin((A - B + C)/2) is equal to c² + a² - b².
- The number of values of x in the interval (0, 5π) satisfying the equation 3sin²x - 7sinx + 2 = 0 is 5.
- If tan⁻¹(1/4) + tan⁻¹(1/2) = cos⁻¹x, then x is 3/√5.
- If cos⁻¹(√12/13) + sin⁻¹(3/5) = sin⁻¹P, then the value of P is 5√3/65.
- In a triangle ABC, I(AB) = √23 units, I(BC) = 3 units, I(CA) = 4 units, then (cot A + cot C) / cot B is 4.
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