Finding Angles - Trigonometry in Right-angled Triangles - Tutorial / Revision (4/5)
Summary
TLDRThis tutorial introduces the use of inverse trigonometric functions to find missing angles in right-angled triangles. It guides viewers through labeling sides, identifying active sides, and applying formulas like tan and cos with their inverses to solve for angles. The instructor demonstrates using a calculator to find angles, rounding the results to 41.0 and 72.2 degrees, respectively. The video concludes with an invitation for questions and a comparison of Pythagoras and trigonometry for future lessons.
Takeaways
- π This tutorial is part of a series on trigonometry, focusing on using inverse trigonometric functions to find missing angles in right-angled triangles.
- π The presenter suggests reviewing previous tutorials if one is not familiar with trigonometric formulas, indicating the importance of foundational knowledge.
- π§ The tutorial introduces inverse trigonometric functions: sinβ»ΒΉ, cosβ»ΒΉ, tanβ»ΒΉ, and how they are accessed on calculators, which is crucial for solving the problems.
- π The first step in solving the problem is labeling the sides of the triangle, which helps in identifying the 'active' sides needed for the calculations.
- π’ The tutorial demonstrates using the tangent function (tan) to find an angle when given the lengths of the opposite and adjacent sides.
- β The use of the inverse tangent function (tanβ»ΒΉ) is shown to isolate and find the value of the angle, with an example calculation provided.
- π The process of substituting values into the trigonometric formula and solving for the angle is explained step by step, emphasizing precision.
- π Another example is given using the cosine function (cos) to find an angle when given the lengths of the adjacent side and the hypotenuse.
- π The inverse cosine function (cosβ»ΒΉ) is used similarly to tanβ»ΒΉ to find the angle, with an example calculation shown.
- π The importance of writing out the formulas in full is highlighted to ensure correct substitution and avoid errors.
- π€ The tutorial ends with an invitation for questions and a reminder of the availability of further resources for understanding trigonometry and its applications.
Q & A
What is the main topic of this tutorial?
-The main topic of this tutorial is using trigonometry to find missing angles in right-angled triangles.
What is the series number of this tutorial in the trigonometry series?
-This is tutorial number 4 in the trigonometry series.
What are the inverse trigonometric functions mentioned in the script?
-The inverse trigonometric functions mentioned are sin^(-1), cos^(-1), and tan^(-1).
How can you access the inverse trigonometric functions on a calculator?
-On a calculator, these functions are usually accessed by pressing shift or sometimes 2nd F and then the sin, cos, or tan buttons.
What are the active sides in a triangle when using trigonometric ratios?
-The active sides are the two sides given in the problem statement, which are used in the trigonometric formula to find the missing angle.
What trigonometric ratio uses the opposite and adjacent sides of a right-angled triangle?
-The tangent ratio (tan) uses the opposite and adjacent sides of a right-angled triangle.
How do you find the angle X using the tangent ratio?
-To find the angle X, you use the inverse tangent function, tan^(-1), on the calculator with the opposite side divided by the adjacent side.
What is the result of the first example calculation in the script?
-The result of the first example calculation is 41.0 degrees for the angle X.
What trigonometric ratio uses the adjacent side and the hypotenuse of a right-angled triangle?
-The cosine ratio (cos) uses the adjacent side and the hypotenuse of a right-angled triangle.
How do you find the angle Y using the cosine ratio?
-To find the angle Y, you use the inverse cosine function, cos^(-1), on the calculator with the adjacent side divided by the hypotenuse.
What is the result of the second example calculation in the script?
-The result of the second example calculation is 72.2 degrees for the angle Y.
What is the purpose of the tutorial mentioned at the end of the script?
-The purpose of the mentioned tutorial is to compare Pythagoras' theorem with trigonometry and to determine when to use each.
Outlines
π Introduction to Trigonometry Tutorial
This paragraph introduces the trigonometry tutorial focused on finding missing angles in right-angled triangles using inverse trigonometric functions. It is the fourth tutorial in a series and assumes familiarity with basic trigonometric formulas. The speaker encourages viewers to catch up on previous lessons and mentions the use of calculator functions like sin^-1, cos^-1, and tan^-1 for solving these problems. The paragraph sets the stage for a step-by-step approach to solving trigonometric problems.
π Understanding and Labeling Sides in Triangles
This paragraph delves into the process of identifying and labeling the sides of a right-angled triangle, which are essential for applying trigonometric formulas. The speaker explains the importance of recognizing 'active sides' and provides a step-by-step guide on how to proceed with the problem-solving process. It emphasizes the need to label sides correctly before selecting the appropriate trigonometric formula to find the missing angle.
π Applying Trigonometric Formulas to Find Angles
The speaker illustrates the application of trigonometric formulas to find missing angles, specifically using the tangent function in this example. The paragraph explains how to set up the formula with given side lengths, substitute the values, and use the inverse tangent function to solve for the angle. It provides a clear demonstration of how to calculate the angle using a calculator and emphasizes the importance of entering values correctly to avoid errors.
π§ Using Cosine to Solve for Another Angle
This paragraph continues the tutorial by showing how to use the cosine function to find another angle in a right-angled triangle. The process involves identifying the adjacent and hypotenuse sides, writing out the cosine formula, and then using the inverse cosine function to solve for the angle. The speaker provides a practical example, demonstrating the calculation and rounding off the result to find the angle's measure.
π Conclusion and Further Resources on Trigonometry
The final paragraph wraps up the tutorial by summarizing the essential aspects of right-angle trigonometry covered in the video. It mentions an upcoming tutorial comparing Pythagorean theorem with trigonometry and when to use each. The speaker invites viewers to ask questions or comment on the YouTube channel or website, promising to assist with any difficulties, and concludes the tutorial with thanks and farewell.
Mindmap
Keywords
π‘Trigonometry
π‘Right-Angled Triangles
π‘Inverse Trigonometric Functions
π‘Sine (sin)
π‘Cosine (cos)
π‘Tangent (tan)
π‘Calculator
π‘Active Sides
π‘Formula
π‘Hypotenuse
π‘Pythagoras
Highlights
Introduction to the tutorial on using trigonometry to find missing angles in right-angled triangles.
This is the fourth tutorial in the series on trigonometry.
Encouragement to review previous tutorials for a refresher on trigonometric formulas.
Explanation of the use of inverse trigonometric functions like sin^-1, cos^-1, and tan^-1.
Instructions on how to access inverse trig functions on calculators.
The importance of knowing where to find the inverse trig functions on various devices.
Step-by-step guide on labeling sides in a triangle for trigonometry problems.
Identification of active sides in a triangle for solving trigonometry problems.
Using the tangent function (tan) when given two side lengths to find an angle.
Substitution of values into the tangent formula to solve for the angle.
Application of the inverse tangent function (tan^-1) to find the angle.
Demonstration of entering values on a calculator to find the angle using tan^-1.
Example calculation resulting in an angle of 41.0 degrees.
Introduction of another example using the cosine function (cos).
Explanation of using the inverse cosine function (cos^-1) to find an angle.
Example calculation using cos^-1 resulting in an angle of 72.2 degrees.
Conclusion of the essential parts of right-angle trigonometry covered in the tutorial.
Invitation for questions and comments on YouTube or the website for further assistance.
Closing remarks and thanks for watching the tutorial.
Transcripts
hi again and welcome to this me versus
math tutorial which is about using
trigonometry to find missing angles in a
right angled triangles this is tutorial
4 in our series on trig so if you've got
some catching up to do first then please
do that otherwise let's get this show on
the road hopefully the trig formulas are
starting to look really familiar to you
by now in fact you're probably getting a
bit sick of them but hey you've probably
started to memorize them so it's all
good just before we get going properly
for this tutorial we're going to be
using what are known as the inverse trig
functions
these look like sin minus 1 cos minus 1
tan minus 1 and so on on a calculator
these are normally accessed by pressing
shift or sometimes 2nd F and then the
sin cos or tan buttons it's often
similar if you're using a calculator on
your phone tablet or other device - or
sometimes they can actually have their
own button on those sorts of devices so
check that you know where to find those
3 functions this question then is asking
us to find a missing angle apart from
that fact it looks fairly similar to the
sorts of questions we were looking at in
a previous tutorial and most of our
actions are actually going to be exactly
the same as they were when we find an
aside so step one as previously label
your sides first now the next step is to
identify what I've been referring to is
the active sides in our triangle
previously one of these was a side which
we were given but one was the side that
we were looking for this time we have
been given two side lengths so these are
our two active sites let's take them so
we're looking for the formula that uses
the opposite and adjacent which is Toa
still just as before again let's now
write our formula out in full
well let's substitute our values in two
we don't know the angle but it is called
X in our question so
gonna replace the theta with an X now
always be really careful that you get
the next two numbers the right way
around the opposite should be on top so
it's 8.6
divided by the adjacent 9.9 now we have
tan x and we just want X so what we
actually have to do now is to cancel out
10 to do this we have to use the inverse
of tan or tan -1 which I spoke about
previously finding on your calculator so
X is actually equal to inverse tan of
8.6 divided by 9.9 again this is the
value of a good dedicated calculator if
you've got one you can enter everything
on the right-hand side now in one go you
use the fraction button if you wish and
just as before you'll get an answer we
need to round it off ultimately we get
41 point 0 degrees let's look at just
one more example as before let's label
our sides identify the two active sides
and as we know 4 angles this is the two
sides we've been given we're going to be
using cars as that uses the adjacent and
the hypotenuse always write the formula
out in full it just helps make sure
everything is substituted into the right
place so cos theta is equal to adjacent
over hypotenuse so cos y is equal to
three point eight over twelve point four
the adjacent on the top the hypotenuse
on the bottom to get Y on its own we
have to get rid of that cause so we need
to use the inverse of cars or cos minus
1 so Y is going to be equal to inverse
cos of three point eight over twelve
point four that's before get a good
calculator where you can enter all this
in one go get an answer and round it up
to give 72 point two degrees
okay that's angles done and that is all
the essential parts really of right
angle trigonometry we've got another
tutorial where we compare Pythagoras in
trigonometry and look at which to use
and when any questions on anything we've
done then as always just supposed to
comment either on YouTube or at the
website at me versus mass comm and I'll
do my best to help as always thanks for
watching take care and bye for now
you
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