How to Use Right Angled Trigonometry
Set up a right triangle model., Identify the known parts of the triangle., Set up a trigonometry equation., Solve for your unknown value., Interpret and report your result., Solve another problem for practice.
Step-by-Step Guide
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Step 1: Set up a right triangle model.
Trigonometry functions can be used to model real world situations involving lengths and angles.
The first step is to define the situation with a right triangle model.For example, suppose you have the following problem:
You are climbing a hill.
You know that the peak of the hill is 500 meters above the base, and you know that the angle of the climb is 15 degrees.
How far must you walk to reach the top? Sketch a right triangle and label the parts.
The vertical leg is the height of the hill.
The top of that leg represents the peak of the hill.
The angled side of the triangle, the hypotenuse, is the climbing trail. -
Step 2: Identify the known parts of the triangle.
When you have your sketch and have labeled the parts of it, you need to assign the values that you know.
On the problem of the hill, you are told that the vertical height is 500 meters.
Mark the vertical leg of the triangle 500 m.
You are told that the climbing angle is 15 degrees.
This is the angle between the base (bottom leg) of the triangle and the hypotenuse.
You are asked to find the distance of the climb, which is the length of the hypotenuse of the triangle.
Mark this unknown as x{\displaystyle x}. , Review the information that you know and what you are trying to learn, and choose the trigonometry function that links those together.
For example, the sine function links an angle, its opposite side and the hypotenuse.
The cosine function links an angle, its adjacent side and the hypotenuse.
The tangent function links the two legs without the hypotenuse.
In the problem with the hill climb, you should recognize that you know the base angle and the vertical height of the triangle, so this should let you know that you will be using the sine function.
Set up the problem as follows:sinθ=oppositehypotenuse{\displaystyle \sin \theta ={\frac {\text{opposite}}{\text{hypotenuse}}}} sin15=500hypotenuse{\displaystyle \sin 15={\frac {500}{\text{hypotenuse}}}} , Use basic algebraic manipulation to rearrange the equation to solve for the unknown value.
You will then use either a table of trigonometric values or a calculator to find the value of the sine of the angle that you know.To find the length of the hill climb, solve the equation for the length of the hypotenuse. sin15=500hypotenuse{\displaystyle \sin 15={\frac {500}{\text{hypotenuse}}}} hypotenuse=500sin15{\displaystyle {\text{hypotenuse}}={\frac {500}{\sin 15}}} hypotenuse=5000.259{\displaystyle {\text{hypotenuse}}={\frac {500}{0.259}}} hypotenuse=1930{\displaystyle {\text{hypotenuse}}=1930} , With any word problem, getting a numerical answer is not the end of the solution.
You need to report your answer in terms that make sense for the problem, using the proper units.For the hill problem, the solution of 1930 means that the length of the climb is 1930 meters. , Consider one more problem, set up a diagram, and then solve for the unknown length.Read the problem.
Suppose a coal bed under your property is at a 12 degree angle and comes to the surface 6 kilometers away.
How deep do you have to dig straight down to reach the coal under your property? Set up a diagram.
This problem actually sets up an inverted right triangle.
The horizontal base represents the ground level.
The vertical leg represents the depth under your property, and the hypotenuse is the 12 degree angle that slopes down to the coal bed.
Label the known and unknown values.
You know that the horizontal leg is 6 kilometers (3.7 mi), and the angle measurement is 12 degrees.
You want to solve the length of the vertical leg.
Set up a trigonometry equation.
In this case, the unknown value that you want to solve is the vertical leg, and you know the horizontal leg.
The trigonometry function that uses the two legs is the tangent. tanθ=oppositeadjacent{\displaystyle \tan \theta ={\frac {\text{opposite}}{\text{adjacent}}}} tan12=opposite6{\displaystyle \tan 12={\frac {\text{opposite}}{6}}} Solve for the unknown value. opposite=tan12∗6{\displaystyle {\text{opposite}}=\tan 12*6} opposite=0.213∗6{\displaystyle {\text{opposite}}=0.213*6} opposite=1.278{\displaystyle {\text{opposite}}=1.278} Interpret your result.
The lengths in this problem are in units of kilometers.
Therefore, your answer is
1.278 kilometers (0.794 mi).
The answer to the question is that you must dig
1.278 kilometers (0.794 mi) straight down to reach the coal bed. -
Step 3: Set up a trigonometry equation.
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Step 4: Solve for your unknown value.
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Step 5: Interpret and report your result.
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Step 6: Solve another problem for practice.
Detailed Guide
Trigonometry functions can be used to model real world situations involving lengths and angles.
The first step is to define the situation with a right triangle model.For example, suppose you have the following problem:
You are climbing a hill.
You know that the peak of the hill is 500 meters above the base, and you know that the angle of the climb is 15 degrees.
How far must you walk to reach the top? Sketch a right triangle and label the parts.
The vertical leg is the height of the hill.
The top of that leg represents the peak of the hill.
The angled side of the triangle, the hypotenuse, is the climbing trail.
When you have your sketch and have labeled the parts of it, you need to assign the values that you know.
On the problem of the hill, you are told that the vertical height is 500 meters.
Mark the vertical leg of the triangle 500 m.
You are told that the climbing angle is 15 degrees.
This is the angle between the base (bottom leg) of the triangle and the hypotenuse.
You are asked to find the distance of the climb, which is the length of the hypotenuse of the triangle.
Mark this unknown as x{\displaystyle x}. , Review the information that you know and what you are trying to learn, and choose the trigonometry function that links those together.
For example, the sine function links an angle, its opposite side and the hypotenuse.
The cosine function links an angle, its adjacent side and the hypotenuse.
The tangent function links the two legs without the hypotenuse.
In the problem with the hill climb, you should recognize that you know the base angle and the vertical height of the triangle, so this should let you know that you will be using the sine function.
Set up the problem as follows:sinθ=oppositehypotenuse{\displaystyle \sin \theta ={\frac {\text{opposite}}{\text{hypotenuse}}}} sin15=500hypotenuse{\displaystyle \sin 15={\frac {500}{\text{hypotenuse}}}} , Use basic algebraic manipulation to rearrange the equation to solve for the unknown value.
You will then use either a table of trigonometric values or a calculator to find the value of the sine of the angle that you know.To find the length of the hill climb, solve the equation for the length of the hypotenuse. sin15=500hypotenuse{\displaystyle \sin 15={\frac {500}{\text{hypotenuse}}}} hypotenuse=500sin15{\displaystyle {\text{hypotenuse}}={\frac {500}{\sin 15}}} hypotenuse=5000.259{\displaystyle {\text{hypotenuse}}={\frac {500}{0.259}}} hypotenuse=1930{\displaystyle {\text{hypotenuse}}=1930} , With any word problem, getting a numerical answer is not the end of the solution.
You need to report your answer in terms that make sense for the problem, using the proper units.For the hill problem, the solution of 1930 means that the length of the climb is 1930 meters. , Consider one more problem, set up a diagram, and then solve for the unknown length.Read the problem.
Suppose a coal bed under your property is at a 12 degree angle and comes to the surface 6 kilometers away.
How deep do you have to dig straight down to reach the coal under your property? Set up a diagram.
This problem actually sets up an inverted right triangle.
The horizontal base represents the ground level.
The vertical leg represents the depth under your property, and the hypotenuse is the 12 degree angle that slopes down to the coal bed.
Label the known and unknown values.
You know that the horizontal leg is 6 kilometers (3.7 mi), and the angle measurement is 12 degrees.
You want to solve the length of the vertical leg.
Set up a trigonometry equation.
In this case, the unknown value that you want to solve is the vertical leg, and you know the horizontal leg.
The trigonometry function that uses the two legs is the tangent. tanθ=oppositeadjacent{\displaystyle \tan \theta ={\frac {\text{opposite}}{\text{adjacent}}}} tan12=opposite6{\displaystyle \tan 12={\frac {\text{opposite}}{6}}} Solve for the unknown value. opposite=tan12∗6{\displaystyle {\text{opposite}}=\tan 12*6} opposite=0.213∗6{\displaystyle {\text{opposite}}=0.213*6} opposite=1.278{\displaystyle {\text{opposite}}=1.278} Interpret your result.
The lengths in this problem are in units of kilometers.
Therefore, your answer is
1.278 kilometers (0.794 mi).
The answer to the question is that you must dig
1.278 kilometers (0.794 mi) straight down to reach the coal bed.
About the Author
Christina Reed
Experienced content creator specializing in hobbies guides and tutorials.
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