Unit 7: Right Triangle Trigonometry
The Pythagorean theorem proved a dozen different ways, the two special right triangles worth memorizing, and the ratios that unlock any right triangle.
10 topics · 34 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Pythagorean Theorem
- How it worksHow do the squares built on each of a right triangle's three sides relate to each other in area?
- Worked exampleFind the length of the hypotenuse of a right triangle with legs 9 and 12.
- Worked exampleFind the missing leg of a right triangle with hypotenuse 17 and one leg 8.
- Common mistakeWhy doesn't the Pythagorean theorem apply to a triangle that isn't a right triangle?
Your turn1 of 2
Find the length of the hypotenuse of a right triangle with legs 15 and 14.Multiple Proofs of the Pythagorean Theorem
- How it worksHow does rearranging four copies of a right triangle inside a square reveal the Pythagorean theorem visually?
- How it worksHow does President Garfield's trapezoid proof use only the area of a trapezoid to reach the same result?
- Worked exampleFind the area of the large square in a Pythagorean rearrangement proof for a right triangle with legs 3 and 4.
- Ties togetherHow does the similar-triangles proof of the Pythagorean theorem end at the same equation as the rearrangement proofs?
Your turn1 of 2
In a Pythagorean rearrangement proof, a right triangle has legs 11 and 10. Find the area of the large square built on the hypotenuse.Special Right Triangles: 45-45-90 and 30-60-90
- How it worksHow does bisecting an equilateral triangle produce the 1, root 3, 2 side ratio of a 30-60-90 triangle?
- Worked exampleFind the two legs of a 45-45-90 triangle with a hypotenuse of 10.
- Worked exampleFind the longer leg and hypotenuse of a 30-60-90 triangle with a shorter leg of 5.
- Common mistakeWhy doesn't doubling one side of a 30-60-90 triangle double all three of its sides equally?
Your turn1 of 2
A 45-45-90 triangle has a hypotenuse of 28. Find the length of each leg.Sine, Cosine, and Tangent Ratios
- How it worksHow do sine, cosine, and tangent each pair up two of a right triangle's three sides with one angle?
- Worked exampleFind sine, cosine, and tangent of the angle opposite the leg of length 4 in a 3-4-5 right triangle.
- Common mistakeWhy isn't the tangent of an angle just the same thing as the angle's sine divided by two?
Your turn1 of 2
In a right triangle, the side opposite an acute angle measures 8 and the hypotenuse measures 17. Find the sine of that angle.Classifying Triangles with the Converse of the Pythagorean Theorem
- How it worksHow does comparing the sum of the squares of the two shorter sides to the square of the longest side classify the triangle?
- Worked exampleFind whether a triangle with sides 6, 8, and 11 is acute, right, or obtuse.
- Common mistakeWhy doesn't the converse of the Pythagorean theorem require you to know any of the triangle's angles first?
Your turn1 of 2
A triangle has sides 8, 8, and 11, where 11 is the longest side. Find the value of a squared plus b squared minus c squared, which classifies the triangle (positive means acute, zero means right, negative means obtuse).Solving for a Missing Side Using Trig Ratios
- How it worksHow do you decide whether to set up sine, cosine, or tangent when a side is missing?
- Worked exampleFind the length of a ladder leaning at 70 degrees from the ground that reaches 12 feet up a wall.
- Worked exampleFind the horizontal distance corresponding to a 40 degree angle of depression looking down at an object 50 feet below.
Your turn1 of 2
A ladder leans against a wall, making an angle of 62 degrees with the ground, and reaches 21 feet up the wall. Find the length of the ladder.Solving for a Missing Angle Using Inverse Trig
- How it worksHow does an inverse trig function undo the ratio to recover the original angle?
- Worked exampleFind the angle whose tangent equals the ratio of a 7 foot rise over a 24 foot run.
- Common mistakeWhy doesn't arcsine of a ratio greater than 1 ever give you a real angle?
Your turn1 of 2
Find the angle, in degrees, whose tangent equals the ratio of a 21 foot rise over a 20 foot run.Angles of Elevation and Depression
- How it worksHow does an angle of elevation from the ground always equal the angle of depression from above?
- Worked exampleFind the height of a building if the angle of elevation to its top is 35 degrees from 100 feet away.
- Common mistakeWhy isn't the angle of elevation measured from the vertical instead of from the horizontal?
Your turn1 of 2
The angle of elevation to the top of a building is 59 degrees, measured from a point 137 feet away on the ground. Find the height of the building.The Law of Sines
- How it worksHow does dropping an altitude in any triangle prove the law of sines connects all three side-angle pairs?
- Worked exampleFind the missing side of a triangle with angles 40 and 65 degrees and the side opposite the 40 degree angle equal to 8.
- Going furtherWhat happens to the law of sines calculation when two different triangles both fit the same given information?
Your turn1 of 2
A triangle has angles of 62 degrees and 59 degrees. The side opposite the 62 degree angle measures 19. Find the length of the side opposite the 59 degree angle.The Law of Cosines
- How it worksHow does the law of cosines reduce to the Pythagorean theorem when the included angle is 90 degrees?
- Worked exampleFind the third side of a triangle with sides 7 and 10 and an included angle of 60 degrees.
- Worked exampleFind the largest angle of a triangle with sides 5, 6, and 10 using the law of cosines.
- Common mistakeWhy can't the law of sines alone solve a triangle when you only know all three side lengths?
Your turn1 of 2
A triangle has sides of length 11 and 11 with an included angle of 159 degrees. Find the length of the third side.