Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall,
Let’s derive the sum formula for tangent.
We can derive the difference formula for tangent in a similar way.
Sum and difference formulas for tangent
The
sum and difference formulas for tangent are:
Given two angles, find the tangent of the sum of the angles.
Write the sum formula for tangent.
Substitute the given angles into the formula.
Simplify.
Finding the exact value of an expression involving tangent
Find the exact value of
Let’s first write the sum formula for tangent and then substitute the given angles into the formula.
Next, we determine the individual function values within the formula:
We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.
To find
we begin with
and
The side opposite
has length 3, the hypotenuse has length 5, and
is in the first quadrant. See
[link] . Using the Pythagorean Theorem, we can find the length of side
Since
and
the side adjacent to
is
the hypotenuse is 13, and
is in the third quadrant. See
[link] . Again, using the Pythagorean Theorem, we have
Since
is in the third quadrant,
The next step is finding the cosine of
and the sine of
The cosine of
is the adjacent side over the hypotenuse. We can find it from the triangle in
[link] :
We can also find the sine of
from the triangle in
[link] , as opposite side over the hypotenuse:
Now we are ready to evaluate
We can find
in a similar manner. We substitute the values according to the formula.
For
if
and
then
If
and
then
Then,
To find
we have the values we need. We can substitute them in and evaluate.
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?