We can find an unknown angle in a right-angled triangle, as long as we know the lengths of two of its sides.
ExampleThe ladder leans against a wall as shown. What is the angle between the ladder and the wall? The answer is to use Sine, Cosine or Tangent! But which one to use? We have a special phrase "SOHCAHTOA" to help us, and we use it like this:
Step 1: find the names of the two sides we know
Example: in our ladder example we know the length of:
Step 2: now use the first letters of those two sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" to find which one of Sine, Cosine or Tangent to use:
In our example that is Opposite and Hypotenuse, and that gives us “SOHcahtoa”, which tells us we need to use Sine.
Step 3: Put our values into the Sine equation: Sin (x) = Opposite / Hypotenuse = 2.5 / 5 = 0.5
Step 4: Now solve that equation! sin(x) = 0.5 Next (trust me for the moment) we can re-arrange that into this: x = sin-1(0.5) And then get our calculator, key in 0.5 and use the sin-1 button to get the answer: x = 30° And we have our answer!
Well, the Sine function "sin" takes an angle and gives us the ratio "opposite/hypotenuse",
But sin-1 (called "inverse sine") goes the other way ... Example:
On your calculator, try using sin and sin-1 to see what results you get! Also try cos and cos-1. And tan and tan-1. Step By StepThese are the four steps we need to follow:
ExamplesLet’s look at a couple more examples:
ExampleFind the angle of elevation of the plane from point A on the ground.
Tan x° = opposite/adjacent = 300/400 = 0.75 tan-1 of 0.75 = 36.9° (correct to 1 decimal place) Unless you’re told otherwise, angles are usually rounded to one place of decimals.
ExampleFind the size of angle a°
cos a° = 6,750/8,100 = 0.8333 cos-1 of 0.8333 = 33.6° (to 1 decimal place) 250, 1500, 1501, 1502, 251, 1503, 2349, 2350, 2351, 3934 Copyright © 2021 MathsIsFun.com
Find angle C. Possible Answers:
Explanation: First, know that all the angles in a triangle add up to 180 degrees. Each triangle has 3 angles. Thus, we have the sum of three angles as shown:
where we have angles A, B, and C. In our right triangle, one angle is 25 degree and we'll call that angle A. The other known angle is 90 degrees and we'll call this angle B. Thus, we have Simplify and solve for C.
Which of the following can be two angle measures of a right triangle? Possible Answers:
Correct answer: Explanation: A right triangle cannot have an obtuse angle; this eliminates the choice of 100 and 10. The acute angles of a right triangle must total 90 degrees. Three choices can be eliminated by this criterion: The remaining choice is correct:
A right triangle has an angle that is 15 more than twice the other. What is the smaller angle? Possible Answers:
Correct answer: Explanation: The sum of the angles in a triangle is 180. A right triangle has one angle of 90. Thus, the sum of the other two angles will be 90. Let = first angle and = second angleSo the equation to solve becomes orThus, the first angle is and the second angle is .So the smaller angle is
Angle in the triangle shown below (not to scale) is 35 degrees. What is angle ?Possible Answers:
degrees degrees degrees degrees
Correct answer: degrees Explanation: The interior angles of a triangle always add up to 180 degrees. We are given angle and since this is indicated to be a right triangle we know angle is equal to 90 degrees. Thus we know 2 of the 3 and can determine the third angle.Angle is equal to 55 degrees.
Which of the following cannot be true of a right triangle? Possible Answers:
One leg can be longer than the hypotenuse.
A right triangle can have an obtuse angle.
A right triangle can be equilateral.
None of the other statements can be true of a right triangle.
The measures of the angles of a right triangle can total .
Correct answer: None of the other statements can be true of a right triangle. Explanation: All of these statements are false. A right triangle can be equilateral. False: An equilateral triangle must have three angles that measure each.One leg can be longer than the hypotenuse. False: Each leg is shorter than the hypotenuse. A right triangle can have an obtuse angle. False: Both angles of a right triangle that are not right must be acute. The measures of the angles of a right triangle can total .False: The measures of any triangle total .
In triangle , what is the measure of angle ?Possible Answers:
Correct answer: Explanation: The formula to find all the angles of a triangle is: To solve for the measure of angle , we plug in the values of and . Since angle is a right angle, we know the measure will be .
Find the degree measure of the missing angle. Possible Answers:
Correct answer:
Explanation: All the angles in a triangle add up to 180º. To find the value of the remaining angle, subtract the known angles from 180º:
Therefore, the third angle measures 43º.
The right triangle has two equal angles, what is each of their measures?Possible Answers:
Correct answer: Explanation: The internal angles of a triangle always add up to 180 degrees, and it was given that the triangle was right, meaning that one of the angles measures 90 degrees. This leaves 90 degrees to split evenly between the two remaining angles as was shown in the question. Therefore, each of the two equal angles has a measure of 45 degrees.
What is the missing angle in this right triangle? Possible Answers:
Correct answer: Explanation: The angles of a triangle all add up to .This means that .Using the fact that 90 is half of 180, we can figure out that the missing angle, x, plus 34 adds to the remaining 90, and we can just subtract
Solve for :Possible Answers:
Correct answer: Explanation: The angles of a triangle add together to 180 degrees. We already know that one of the angles is 90 degrees, so we can subtract 90 from 180: the other 2 angles have to add to 90 degrees. We can now subtract to get x:
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