What Is Tangent Of An Angle

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What Is Tangent Of An Angle – Any high school student who has encountered trigonometry in his Mathematics curriculum has probably encountered sine, cosine and area rules commonly used to solve triangles where certain information is given and the rest is to be calculated. Somewhat surprisingly (because it is relatively simple to derive), the “tanning rule” is generally not included as part of this particular set of trig tools. However, as we hope to show in this article, this rule can be very useful in some cases. Especially in the case where two sides and an included angle of a triangle are given. In this case, it is impossible to immediately apply the sine rule to determine the remaining angles, since none of the given sides is opposite the given angle. The cosine rule must first be applied to determine the third side and then the sine rule for one (or both) of the remaining angles. However, the tan rule allows you to determine one of the two missing angles without resorting to either the sine rule or the cosine rule.

The following production follows the steps of the one presented in Appendix A. The last one is a translation (from Stephen Hawking’s book: On the Shoulders of Giants ) of a passage from the “Liber Primus” of “De Revolutionibus Orbium Coelestium” written by a Nicolaus Copernicus and published posthumously in 1543. It must be appreciated that the “theorem” is older, since Copernicus drew his geometry from Greek sources, especially Euclid and Ptolemy. However, Copernicus notes the subsequent introduction of Arabic numerals:

What Is Tangent Of An Angle

What Is Tangent Of An Angle

“This notation surpasses any other — Greek or Latin — in a certain singular ease of work, and easily adapted to any class of calculation.”

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In the following diagram of triangle ABC, the sides AB (ie c)  and BC (ie a) are given as the included angle ABC ##(hat)##. From the point A draw a perpendicular to the point D of the side BC extending if necessary according to whether D is in ##;(hat## acute) or outside the triangle ##;(hat# # obtuse). The triangle is now divided into two right triangles ABD and ADC. Since the angle at D is a right angle and ##hat## is given, we can calculate AD as ##csinhat## and BD as ##ccoshat##. CD can be determined as the difference between BC and BD: ##a-ccoshat##.

What Is Tangent Of An Angle

(Note that this difference will be negative in the case that ##hat## is obtuse, so ##tanhat## will be negative as expected).

Therefore, in the right triangle ADC, the sides AD and CD are given from which we can determine AC  and the angle ACD.

What Is Tangent Of An Angle

Handout K1: Angles Formed By Tangents

And it will not prove otherwise if the angle B is obtuse as in the diagram above. The perpendicular dropped from point A to the extended line BC creates a right triangle ABD in which AD=c sin(180-##hat##)=c sin##hat## and BD=c cos(180- ## hat##)=-c cos##hat##.

Since BA and BC have a given ratio to each other (BA:BC = c:a = k:1), therefore AB is given (c=ka) in the same places where BD (##=-ka coshat ##) and the entire CBD is given (##=a-kacoshat##).

What Is Tangent Of An Angle

They correspond in the right triangle ADC, since the two sides, AD and CD, are given, AC (ie b) the intended side and the angle ##hat## are also given.

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An important point to note in the second half (##hat## obtuse) of the above production is that Copernicus switches from referring to fixed sides to a fixed aspect ratio. To solve the angle C with the “tanning rule”, the absolute values ​​of the pair of given sides are not necessary, as long as we know the ratio between them. This is reflected in the final formula for ##tanhat## above and illustrated in working examples 1 and 4.

What Is Tangent Of An Angle

The user PF Kuruman notes that once the formula for (say) ##tanhat=frac}}}}## is given, the other two tan ratios can be found by circular transposition, # #Arightarrow B rightarrow C rightarrow A##  of lowercase and uppercase letters. Or we start from ##tanhat=frac}}}}## with a similar circular permutation in the opposite direction ##Arightarrow C rightarrow B rightarrow A##.

We shall consider three physical applications in which the “tanning rule” may be usefully employed. The first is in Compton scattering in which a photon is incident on a stationary electron and there is a transfer of momentum that results in the photon being “expected” at an angle that depends on the change in frequency (energy) of the incident and scattered photons. Usually there are two angles to be determined, namely that of the scattered photon and that of the electron. Both are measured from the initial direction of motion of the incident photon.

What Is Tangent Of An Angle

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The second is in determining the angle (s) of projection of a missile if it is to hit a target for which horizontal and vertical coordinates (measured from the point of projection) are known. In this application we will use a vector equation ##vectimesvec=vectimesvec## which was described in a previous article entitled “Fourth Order in Projectile Motion”.

The third is in two-dimensional elastic collisions between two masses – one moving and one stationary. As in Compton scattering, we find that the post-collision trajectory angles (relative to the pre-collision direction of the moving mass) of the two masses depend on the energy transferred from one to the other and (especially in this case) the reason for the masses. And having determined one angle from an energy relationship, we can determine the other using the tan rule.

What Is Tangent Of An Angle

It will be seen that there are many interesting similarities between these apparently different physical applications. Momentum and/or thrust are physical vector quantities in any state, and the angle of attack is related to the change in energy. However, the 3 vector quantities involved form a triangle in which 2 sides and an included angle are known or can be calculated. A second angle can be determined by applying the “tanning rule”.

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While in this article we focus our attention on applications involving momentum and energy, the “tanning rule” is certainly as versatile in use as its “brothers”, the sine rule and the cosine rule . In the first of the “worked examples” presented, the “tanning rule” is illustrated in a problem involving 3 forces in equilibrium. Clearly, the vector sum of such forces will be zero, and the associated vector diagram is a closed triangle in which the ratio of two sides is given with the included angle, and we must determine a pair of unknown angles. The other 3 worked examples relate to the physical applications mentioned above.

What Is Tangent Of An Angle

A vector diagram reflecting Compton scattering is shown below. The momentum of the incident photon is equal to the sum of the moments of the scattered photon and the incident electron. The electron is assumed to be initially at rest.

To apply the Tanning rule to determine the orbital angle ##phi## of the electron, we first need the deflection angle ##theta_d## between the incident and the scattered photon. We may just use a known result, but for completeness, we will include a short derivation of it.

What Is Tangent Of An Angle

Answered: 3. What Is The Tangent Ratio Of Angle…

As mentioned above, we will use the vector equation ##vectimesvec=vectimesvec## to determine the two possible launch angles for a missile aimed at a target of known coordinates (such as measured from launch). ). ##vec a## is the acceleration, ##vec## displacement from the launch point  and ##vec u## and ##vec v## the initial and final velocity of the projectile respectively. In the case of projectile motion ###vec a=0;hat} -g;hat}## and ###vec s=R;hat} + h;hat}# #, where R is the target’s horizontal distance and h its vertical height above the launch point. Therefore,  ##vectimesvec=gR ;hat}##. We can write ##vectimesvec## as ##|v| |u| sintheta_d; hat}## where ###theta_d## is the angle between the initial and final velocity vectors. From this we have the equation: $$sintheta_d=frac.$$ Here we followed the method described by J. Gibson Winans in an article entitled “Convenient Equations for Projectile Motion” from the American Journal of Physics. In this paper, a construction technique was used to determine launch angles, while here we adopted an analytical approach that uses the “tanning rule” described above. Also in this article, the aforementioned vector equation was derived from the mathematics of quadratic products, while here we present a derivation based on Newton’s second law applied to angular momentum.

We start by “converting” the vector equation into an angular momentum equation by the simple solution of multiplying both sides by the mass m of the projectile: $$mvectimesvec=mvectimesvec. $$ So, in the context of projectile motion, ##mvectimesvec## represents the gravitational moment of the projectile relative to its launch point. Taking our cue from PF user @TSny (who originally identified the physical meaning of vector equations as described here), we can show that the right-hand side of the equation ##(mvec v times vec u)# # represents the rate of change of angular momentum according to Newton’s second law applied to angular motion: $$L=mvectimesvec=mvectimesfrac+vec )t}=fractimesvec )t}$$

What Is Tangent Of An Angle

$$indicate dot L=mleft(fractimesvec}+fractimesvec)t}right).$$ Write ##vec a## as ##frac# # , the above simplifies to $$dot L =mleft(fractimesvec}+fractimesvec}right)=mvec v times vec u$$

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Using the “tanning rule,” we can now “solve” the following velocity vector diagram to obtain an expression for the launch angles in terms of the projectile’s initial velocity, horizontally.

What Is Tangent Of An Angle

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