Podcast
Questions and Answers
Given the equation $\sqrt{3} \tan(2\theta) + \sqrt{3} \tan(3\theta) + \tan(2\theta)\tan(3\theta) = 1$, what is the general value of $\theta$?
Given the equation $\sqrt{3} \tan(2\theta) + \sqrt{3} \tan(3\theta) + \tan(2\theta)\tan(3\theta) = 1$, what is the general value of $\theta$?
- $(n + \frac{1}{3})\frac{\pi}{5}$ (correct)
- $n\pi + \frac{\pi}{5}$
- $(n + \frac{1}{6})\frac{\pi}{5}$
- $(2n \pm \frac{1}{6})\frac{\pi}{5}$
Determine $\theta$ if $\tan(\theta) + \tan(2\theta) + \tan(\theta)\tan(2\theta) = 1$.
Determine $\theta$ if $\tan(\theta) + \tan(2\theta) + \tan(\theta)\tan(2\theta) = 1$.
- $\frac{n\pi}{2} + \frac{\pi}{6}$ (correct)
- $\frac{n\pi}{2} + \frac{\pi}{12}$
- $\frac{n\pi}{2} + 6$
- $\frac{n\pi}{3} + \frac{\pi}{12}$
If $2\tan^2(\theta) = \sec^2(\theta)$, find the general value of $\theta$.
If $2\tan^2(\theta) = \sec^2(\theta)$, find the general value of $\theta$.
- $n\pi - \frac{\pi}{4}$
- $n\pi \pm \frac{\pi}{4}$ (correct)
- $2n\pi \pm \frac{\pi}{4}$
- $n\pi + \frac{\pi}{4}$
What is the general solution of the equation $\tan(\theta)\tan(2\theta) = 1$?
What is the general solution of the equation $\tan(\theta)\tan(2\theta) = 1$?
Given $\cos(\theta) + \cos(7\theta) + \cos(3\theta) + \cos(5\theta) = 0$, find the value of $\theta$.
Given $\cos(\theta) + \cos(7\theta) + \cos(3\theta) + \cos(5\theta) = 0$, find the value of $\theta$.
If $\tan(30) + 1 = \sqrt{3}$, what is the general form of the solution?
If $\tan(30) + 1 = \sqrt{3}$, what is the general form of the solution?
What is the value of $x$ in the equation $\sin(3\alpha) = 4\sin(\alpha)\sin(x + \alpha)\sin(x - \alpha)$?
What is the value of $x$ in the equation $\sin(3\alpha) = 4\sin(\alpha)\sin(x + \alpha)\sin(x - \alpha)$?
Determine the value of $\theta$ that satisfies the equation $\tan(\theta) + \tan(20) + \tan(\theta)\tan(20) = 1$.
Determine the value of $\theta$ that satisfies the equation $\tan(\theta) + \tan(20) + \tan(\theta)\tan(20) = 1$.
Find the general solution for $\theta$ in the equation $2\tan^2(\theta) = \sec^2(\theta)$.
Find the general solution for $\theta$ in the equation $2\tan^2(\theta) = \sec^2(\theta)$.
What is the general solution to the equation $\tan(\theta)\tan(2\theta) = 1$, where $n \in I$?
What is the general solution to the equation $\tan(\theta)\tan(2\theta) = 1$, where $n \in I$?
Determine the value of $x$ if $\sin(3\alpha) = 4\sin(\alpha)\sin(x + \alpha)\sin(x - \alpha)$.
Determine the value of $x$ if $\sin(3\alpha) = 4\sin(\alpha)\sin(x + \alpha)\sin(x - \alpha)$.
If $\cos(\theta) + \cos(7\theta) + \cos(3\theta) + \cos(5\theta) = 0$, then what is $\theta$?
If $\cos(\theta) + \cos(7\theta) + \cos(3\theta) + \cos(5\theta) = 0$, then what is $\theta$?
If $\sqrt{3} \tan{20} + \sqrt{3} \tan{30} + \tan{20} \tan{30} = 1$, find the general value of $\theta$.
If $\sqrt{3} \tan{20} + \sqrt{3} \tan{30} + \tan{20} \tan{30} = 1$, find the general value of $\theta$.
Determine the value of $\theta$ satisfying $\tan(\theta) + \tan(20) + \tan(\theta)\tan(20) = 1$.
Determine the value of $\theta$ satisfying $\tan(\theta) + \tan(20) + \tan(\theta)\tan(20) = 1$.
Solve for the general value of $\theta$ given $2\tan^2(\theta) = \sec^2(\theta)$.
Solve for the general value of $\theta$ given $2\tan^2(\theta) = \sec^2(\theta)$.
Given $\tan(\theta)\tan(2\theta) = 1$, identify the general solution.
Given $\tan(\theta)\tan(2\theta) = 1$, identify the general solution.
If $\sin(3\alpha) = 4\sin(\alpha)\sin(x + \alpha)\sin(x - \alpha)$, find $x$.
If $\sin(3\alpha) = 4\sin(\alpha)\sin(x + \alpha)\sin(x - \alpha)$, find $x$.
Given that $\tan(30) + 1 = \sqrt{3}$, what is the general solution?
Given that $\tan(30) + 1 = \sqrt{3}$, what is the general solution?
Flashcards
General solution of ( \tan \theta \tan 2\theta = 1 )
General solution of ( \tan \theta \tan 2\theta = 1 )
The general solution is ( n\pi \pm \frac{\pi}{6} ), where ( n ) is an integer.
General value of ( \theta ) if ( 2\tan^2 \theta = \sec^2 \theta )
General value of ( \theta ) if ( 2\tan^2 \theta = \sec^2 \theta )
The general value of ( \theta ) is ( n\pi \pm \frac{\pi}{4} ), where ( n ) is an integer.
Value of ( x ) if ( \sin 3\alpha = 4 \sin \alpha \sin (x + \alpha) \sin (x - \alpha) )
Value of ( x ) if ( \sin 3\alpha = 4 \sin \alpha \sin (x + \alpha) \sin (x - \alpha) )
After solving, ( x = n\pi \pm \frac{\pi}{3} ) where ( n ) is an integer.
Study Notes
- Internal residual stresses exist within a material or object without external loads or forces.
- These stresses can be either beneficial or detrimental.
- The impact depends on their magnitude, distribution, and whether they are compressive or tensile.
Origin of Internal Residual Stresses
- Non-uniform plastic deformation: Uneven plastic deformation leads to internal stresses to maintain material compatibility.
- Heat treatments: Processes like quenching and tempering create internal stresses due to temperature differences and phase transformations.
- Phase transformations: Volume changes during transformations (e.g., martensitic in steels) can generate internal stresses.
- Welding: Generates residual stresses from the weld metal's contraction and temperature variations.
- Machining: Induces surface stresses due to plastic deformation from the cutting tool.
- Other processes such as rolling, extrusion, forging and shot peening.
Effects of Internal Residual Stresses
- Fatigue resistance: Compressive surface stresses enhance fatigue resistance, while tensile stresses reduce it.
- Corrosion resistance: Stresses can influence corrosion rates, accelerating it in areas with tensile stress.
- Dimensional stability: Internal stresses may cause instability, leading to deformation over time.
- Fracture resistance: Affects fracture resistance by altering crack initiation probabilities.
- Behavior under load: Stresses impact how a material behaves under load by changing the stress and strain distribution.
Measuring Internal Residual Stresses
- Destructive methods: These methods involve partly or completely destroying the material to measure the internal stresses, like the hole drilling, layer removal, and dissection methods.
- Non-destructive methods: These methods measure the internal stresses non-invasively without destroying the material Some examples are X-ray diffraction, ultrasonic method, and magnetoelastic method.
Methods for Reducing or Redistributing Internal Residual Stresses
- Heat treatments: Annealing and tempering reduce stress by allowing the material to relax.
- Vibration: Reduces stress by facilitating dislocation movement.
- Shot peening: Compressive surface stresses are introduced, which increases resistance to fatigue and corrosion.
- Stretching: Controlled plastic deformation is used to reduce existing stresses.
Examples of Internal Residual Stress Applications
- Surface hardening of steels: Compressive stresses are created on the surface, thus increasing resistance to fatigue and wear.
- Shot peening aircraft components: Increases fatigue resistance and reduces failure.
- Pre-stressed concrete: Introduces compressive stresses, which enhances tensile strength and allows lighter structures to be built.
Summary
- Understanding the source, effects, measurement, and control allows them to be used beneficially which minimizes failures.
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