Right angle trigonometry is useful when dealing with triangles and is an essential part of trigonometry in general. By using the proportions arising from the right angle and understanding the application of goniometric circumference, you can solve a wide variety of problems involving angles and lengths. You develop a system in which you model a problem using a right triangle and then choose the best trigonometric relationship to solve it.
Steps
Method 1 of 3: Using Trigonometric Functions to Measure Distances
Step 1. Lay out the model of a right triangle
You can use trigonometric functions to model real-world situations involving lengths and angles. As a first step, you must define the situation with the model of a right triangle.
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For example, imagine you have the following problem:
- You are climbing up a hill. You know that the top is 500 meters above the base and that the angle of climb is 15 degrees. How far do you have to walk to get to the top?
- Sketch a right triangle and label the parts. The vertical leg is the height of the hill. The upper end of that leg represents the top of the hill. The angled leg of the triangle (the hypotenuse) is the path you climb.
Step 2. Identify the parts you know of the triangle
Once you have the sketch and have labeled its parts, you need to assign the values that you know.
- In the hill problem, you are informed that the vertical height is 500 meters. Mark the vertical leg of the triangle as "500 m."
- You know the angle of climb is 15 degrees. This constitutes the angle between the base (the lower leg) of the triangle and the hypotenuse.
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You are asked to find the distance of the climb; that is, the length of the hypotenuse of the triangle. Mark this unknown as x { displaystyle x}
Step 3. Set up a trigonometric equation
Review the information you know and the information you are trying to find out and choose the trigonometric function that links them. For example, the sine function links an angle, the opposite leg, and the hypotenuse. The cosine function links an angle, the adjacent leg, and the hypotenuse. The tangent function links both legs without the hypotenuse.
- In the hill climb problem, you must recognize that you know the base angle and the vertical height of the triangle. Therefore, you should know that you will use the sinus function. Arrange the problem as follows:
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sinθ = oppositehypotenuse { displaystyle \ sin \ theta = { frac { text {opposite}} { text {hypotenuse}}}}
- sin15=500hipotenusa{displaystyle \sin 15={frac {500}{text{hipotenusa}}}}
Step 4. Find the unknown value
Using basic algebraic manipulation, rearrange the equation to find the unknown value. Next, you will use a table of trigonometric values or a calculator to find the value of the sine of the angle you know.
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You can find the length of the hill climb by solving the equation to find the length of the hypotenuse.
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sin15 = 500hypotenuse { displaystyle \ sin 15 = { frac {500} { text {hypotenuse}}}}
- hipotenusa=500sin15{displaystyle {text{hipotenusa}}={frac {500}{sin 15}}}
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- hipotenusa=5000, 259{displaystyle {text{hipotenusa}}={frac {500}{0, 259}}}
- hipotenusa=1930{displaystyle {text{hipotenusa}}=1930}
Step 5. Interpret and report the result
With any word problem, the solution does not end with getting a numerical answer. You need to report your answer in terms that make sense for the problem with the appropriate units.
In the case of the hill problem, the 1930 solution implies that the length of the climb is 1930 meters
Step 6. Solve another problem to practice
Consider one more problem, draw up a diagram, and then solve to find the unknown length.
- Read the problem. Imagine that a bed of coal under your property is at an angle of 12 degrees and rises to the surface 6 km away. How deep do you have to dig straight down to get to the coal under your property?
- Lay out a diagram. Actually, this problem presents an inverted right triangle. The horizontal base represents the ground level. The vertical leg represents the depth below 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 km (3.7 miles) and the angle measure is 12 degrees. You must find the length of the vertical leg.
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Set up a trigonometric equation. In this case, the unknown value you want to find is the vertical leg, and you already know the horizontal leg. The trigonometric function that uses two legs is the tangent.
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tanθ = oppositeadjacent { displaystyle \ tan \ theta = { frac { text {opposite}} { text {adjacent}}}}
- tan12=opuesto6{displaystyle \tan 12={frac {text{opuesto}}{6}}}
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Encuentra el valor desconocido.
- opuesto=tan12∗6{displaystyle {text{opuesto}}=\tan 12*6}
Método 2 de 3: Usar funciones inversas para calcular ángulos
Step 1. Read the problem with the unknown angle
Trigonometry can also be used to calculate angle measures. Although the process is similar, the problem will ask you to measure an unknown angle.
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Consider the following problem:
At a certain time of day, a 60-meter (200-foot) high flagpole casts a shadow 24 meters (80 feet) long. What is the angle of the sun at this time of day?
Step 2. Sketch a right triangle and label its parts
Don't forget that trigonometry problems are based on the geometry of right triangles. Sketch a right triangle to represent the problem and label the values you know and those you don't.
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In the case of the flagpole problem, the vertical leg is the flagpole itself. Label it with a height of 60 meters (200 feet). The horizontal base of the triangle represents the length of the shadow. Label the base as "24 meters" (80 feet). In this case, the hypotenuse does not represent any physical measurement but the length from the top of the antler to the end of the shadow. This will give you the angle you want to find. Label the angle between the hypotenuse and the base as angle θ { displaystyle \ theta}
Step 3. Set up a trigonometric equation
You need to review the parts of the triangle that you know and those that you need to find. This will help you choose the appropriate trigonometric function to help you find the unknown value.
- In the case of the pole, you know the vertical height and the horizontal base but you don't know the hypotenuse. The function that uses the proportion between the two legs is the tangent.
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Set up an equation for the tangent as follows:
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tanθ = oppositeadjacent { displaystyle \ tan \ theta = { frac { text {opposite}} { text {adjacent}}}}
- tanθ=6024{displaystyle \tan \theta ={frac {60}{24}}}
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- tanθ=2, 5{displaystyle \tan \theta =2, 5}
Step 4. Use the inverse trigonometric function to find the measure of the angle
When you have to find the measure of the angle itself, you need to use what is known as the inverse trigonometric function. Inverse functions are called "arc" functions. They are the arcsine ("arcsin"), the arccosine ("arccos") and the arc tangent ("arctan").
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In a calculator, these functions appear as sin − 1 { displaystyle sin ^ {- 1}}
, cos−1{displaystyle cos^{-1}}
y tan−1{displaystyle tan^{-1}}
. You enter the value and then press the appropriate button to get the angle measure. Some calculators vary, as in some cases you enter the value first and then press the button." arctan"="" y,="" en="" otros,="" presionas="" el="" botón="" de="" "arctan"="" y="" luego="" ingresas="" el="" valor.="" deberás="" determinar="" el="" proceso="" que="" funcione="" para="" tu=" />
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tanθ = 2.5 { displaystyle \ tan \ theta = 2, 5}
- θ=arctan2, 5{displaystyle \theta =\arctan 2, 5}
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- θ=68, 2{displaystyle \theta =68, 2}
Step 5. Interpret the result
The unit of the result will be degrees, since you were going to find the measure of an angle. Check to see if the answer makes sense.
According to this solution, the angle between the Earth and the sun is 68.2 degrees. At noon, the sun is directly overhead, which would be a 90 degree angle. Hence, this solution seems reasonable
Step 6. Set up another problem with an unknown angle
Each time the angle measure is the unknown factor, you will use an inverse trigonometric function. Generally, the procedure is always the same.
- Read the problem. A right triangle with legs that are 7, 5, and 10 cm (3, and 4 inches) long has a hypotenuse that is 12 cm (5 inches) long. What is the measure of the angle opposite the 7.5-cm (3-inch) leg?
- Sketch the problem. In this case, the problem is only related to the measurements of a triangle. Make a sketch of a right triangle and label the information you know. One leg is 7.5 cm, the other 10 cm, and the hypotenuse 12 cm. For this problem, the unknown angle is the acute angle opposite the leg of 7.5 cm (3 inches).
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Set up a trigonometric equation. In this case, because you know all three sides of the triangle, you can actually choose the function. You have the necessary data to use any of the functions (sine, cosine or tangent) in the following way:
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sinθ = oppositehypotenuse { displaystyle \ sin \ theta = { frac { text {opposite}} { text {hypotenuse}}}}
- cosθ=adyacentehipotenusa{displaystyle \cos \theta ={frac {text{adyacente}}{text{hipotenusa}}}}
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- tanθ=opuestoadyacente{displaystyle \tan \theta ={frac {text{opuesto}}{text{adyacente}}}}
Step 7. Plug in the values you know and find the unknown angle
In this case, continue solving with the three functions to eventually realize that the three different functions reach the same conclusion regarding the value of the angle θ { displaystyle \ theta}
- En primer lugar, dispón una solución con la función sin{displaystyle \sin }
- sinθ=opuestohipotenusa{displaystyle \sin \theta ={frac {text{opuesto}}{text{hipotenusa}}}}
- sinθ=7, 512{displaystyle \sin \theta ={frac {7, 5}{12}}}
- sinθ=0, 625{displaystyle \sin \theta =0, 625}
- Luego, dispón una solución con la función cos{displaystyle \cos }
- cosθ=adyacentehipotenusa{displaystyle \cos \theta ={frac {text{adyacente}}{text{hipotenusa}}}}
- cosθ=1012{displaystyle \cos \theta ={frac {10}{12}}}
- cosθ=0, 83{displaystyle \cos \theta =0, 83}
- Por último, dispón una solución con la función tan{displaystyle \tan }
- tanθ=opuestoadyacente{displaystyle \tan \theta ={frac {text{opuesto}}{text{adyacente}}}}
- tanθ=7, 510{displaystyle \tan \theta ={frac {7, 5}{10}}}
- tanθ=0, 75{displaystyle \tan \theta =0, 75}
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Step 8. Find the values of the arc functions with a calculator or trigonometric table to find the measure of the angle
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Find the measure with arcsin { displaystyle \ arcsin}
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- sinθ=0, 625{displaystyle \sin \theta =0, 625}
- θ=arcsin0, 625{displaystyle \theta =\arcsin 0, 625}
- θ=36, 9{displaystyle \theta =36, 9}
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- cosθ=0, 83{displaystyle \cos \theta =0, 83}
- θ=arccos0, 83{displaystyle \theta =\arccos 0, 83}
- θ=36, 9{displaystyle \theta =36, 9}
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- tanθ=0, 75{displaystyle \tan \theta =0, 75}
- θ=arctan0, 75{displaystyle \theta =\arctan 0, 75}
- θ=36, 9{displaystyle \theta =36, 9}
Step 9. Review your results
In this problem, you were able to obtain the solution in three different ways because you started with an angle and the measurements of the three sides. Any one of them alone would have been enough to find the answer. By solving all three, you see that the solution is the same either way. In this case, the chosen angle is 36.9 degrees.
Method 3 of 3: Define the Basic Features
Step 1. Understand the goniometric circumference
Trigonometry is based on the mathematical concept of the goniometric circumference. This constitutes a circle that is drawn on the x - y coordinate plane, with the center being the point (0, 0) and with a radius of 1. By making the radius equal to 1, you can measure the trigonometric functions directly.
- If you visualize a goniometric circle, any point within that circle forms a right triangle. From a selected point on the circle, draw a vertical line directly to the x-axis. Then, from that point on the x-axis, draw a horizontal line to join it to the origin. These two lines, the vertical and the horizontal, will be the legs of a right triangle. The radius of the circle connecting the point on the circle to the center at the origin is the hypotenuse of the right triangle.
- Trigonometric functions still apply for triangles and lengths other than 1, but making the radius equal to 1 makes it more straightforward to calculate the proportions.
Step 2. Learn the breast relationship
The sine function constitutes the proportion between the leg opposite a chosen angle and the hypotenuse of the right triangle. In goniometric circumference, the sine is a way of measuring the vertical distance from the x-axis to the designated point. In other words, it is the y-axis coordinate of the chosen point.
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The sine of an angle is often abbreviated as "sin." The angle of measure is usually labeled θ { displaystyle \ theta}
por convención, por lo que uno dice que mide sinθ{displaystyle \sin \theta }
o sin(θ){displaystyle sin(theta)}
- Por ejemplo, en caso de que elijas un ángulo, llamado θ{displaystyle \theta }
, de 30 grados en el centro de la circunferencia goniométrica, esto marcaría un punto en el círculo cuyas coordenadas serían (32, 12){displaystyle ({frac {sqrt {3}}{2}}, {frac {1}{2}})}
. Luego, puedes decir que sinθ=12{displaystyle \sin \theta ={frac {1}{2}}}
Step 3. Review the cosine function
The cosine function is the ratio between the leg adjacent to the chosen angle and the hypotenuse of the right triangle. On the goniometric circumference, the cosine constitutes the length of the horizontal leg, which is also the coordinate of the point on the circumference on the x-axis.
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The cosine of an angle is often abbreviated as "cos". One says that it measures cosθ { displaystyle \ cos \ theta}
o cos(θ){displaystyle \cos(theta)}
- Por ejemplo, en caso de que elijas un ángulo θ{displaystyle \theta }
de 30 grados en el centro de la circunferencia goniométrica, esto marcaría un punto en la circunferencia cuyas coordenadas serían (32, 12){displaystyle ({frac {sqrt {3}}{2}}, {frac {1}{2}})}
. Luego, puedes decir que cosθ=32{displaystyle \cos \theta ={frac {sqrt {3}}{2}}}
Step 4. Understand the function of the tangent
The third common trigonometric function is the tangent. This constitutes the proportion between the two legs of the right triangle without reference to the hypotenuse. Specifically, for a chosen angle of a right triangle, the tangent is found by dividing the length of the leg opposite the chosen angle by the leg adjacent to the chosen angle. In the goniometric circle, the tangent equals the y-axis coordinate divided by the x-axis coordinate.
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The tangent function is often abbreviated as "tan." In the case of a chosen angle θ { displaystyle \ theta}
, uno dice que mide tanθ{displaystyle \tan \theta }
o tan(θ){displaystyle \tan(theta)}
- En el caso del ejemplo de un ángulo θ{displaystyle \theta }
- tanθ=1232=33=0, 577{displaystyle \tan \theta ={frac {frac {1}{2}}{frac {sqrt {3}}{2}}}={frac {sqrt {3}}{3}}=0, 577}
- Observa que, por lo general, reportar el resultado en términos de una fracción con la raíz cuadrada (por ejemplo, 33{displaystyle {frac {sqrt {3}}{3}}}
de 30 grados en el centro de la circunferencia goniométrica, recuerda que las coordenadas son (32, 12){displaystyle ({frac {sqrt {3}}{2}}, {frac {1}{2}})}
. Puedes encontrar la tangente si divides el seno (la coordenada en el eje y) entre el coseno (la coordenada en el eje x) de la siguiente forma:
) se considera más preciso y exacto que redondear a un decimal (por ejemplo, 0, 577). Para fines prácticos, podrían ser aceptables tres cifras decimales.
Step 5. Review the other proportions
From time to time, you may need alternate proportions to the cosine, sine, and tangent. These alternative functions are the inverses of these first three. While they are not as commonly used in basic calculations, they become essential in more advanced trigonometric work. These functions are as follows:
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Secant: abbreviated as "sec" and equals 1cos { displaystyle { frac {1} {cos}}}
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<li> Cotangent: abbreviated as" cot"="" y="" equivale="" a="" 1tan{displaystyle="" {frac=" />
Step 6. Learn the mnemonic SOHCAHTOA
When trying to remember the proportions of the primary functions sin, cos, and tan, many students employ the "SOHCAHTOA" memorization tool. If divided into its parts, give the proportions as follows:
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SOH are the initials for sine, opposite, and hypotenuse and brings to mind the proportion:
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sin = oppositehypotenuse { displaystyle \ sin = { frac { text {opposite}} { text {hypotenuse}}}}
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cah son las iniciales de coseno, adyacente e hipotenusa de la siguiente forma:
- cos=adyacentehipotenusa{displaystyle \cos ={frac {text{adyacente}}{text{hipotenusa}}}}
toa son las iniciales de tangente, opuesto y adyacente y representa la proporción:
- tan=opuestoadyacente{displaystyle \tan ={frac {text{opuesto}}{text{adyacente}}}}
consejos
- los valores para el seno y el coseno siempre se encuentran entre -1 y 1. sin embargo, la tangente puede equivaler a cualquier número. en caso de que obtengas un error en la función trigonométrica inversa, es probable que el valor sea demasiado grande o demasiado pequeño. revisa la proporción y vuelve a intentarlo. un error común es invertir los lados en la proporción (por ejemplo, dividir la hipotenusa entre el opuesto para el seno).
- sin-1 no es igual a la cosecante, cos-1 no es igual a la secante y tan-1 no es igual a la cotangente. la primera es la función trigonométrica inversa, lo que significa que, en caso de que ingreses el valor de una proporción, obtendrás el ángulo correspondiente, en tanto que la segunda es la función recíproca, lo que quiere decir que la proporción está invertida.