# Between geometry and trigonometry, which is more difficult

## trigonometry

Task 1: Fill in the correct terms below.

Attempts: 0

Trigonometry makes use of the similarity of triangles. If a right triangle has an angle of 30 ° as in the example on the right, then the length ratio between the red and green lines is 1 to 2 (½). So if the red line is 1 cm long, then the green line is 2 cm long. If the red line measures 2 cm, the green line measures 4 cm, etc. In the case of right-angled triangles, the respective aspect ratios can be called up for any angle using a calculator. This makes it possible either to determine the second side length with an angle and a side length or to determine the corresponding angle with two side lengths.

 Example: angle → 30° Aspect ratio (red / green) → ½ given green route → 6 cm searched red route → 6 cm · ½ =? cm

Task 2: Match the color words in the drop-down menu with the correct colors on the sides of the triangle.

### Aspect ratios of sine, cosine and tangent

Task 3: Fill in the missing terms.

Attempts: 0

Info: The aspect ratio of the sides at a corresponding angle is denoted as follows:

• The Sine (short: sin) of an angle is the length ratio of the opposite cathetus of this angle and the hypotenuse.  sin φ = Opposite cathete sin α = a ; sin β = b hypotenuse c c

• The cosine (short: cos) of an angle is the length ratio of the adjacent side of this angle and the hypotenuse.  cos φ = Affiliate cos α = b ; cos β = a hypotenuse c c

• The tangent (short: tan) of an angle is the length ratio of the opposite side of this angle and the adjacent side.  tan φ = Opposite cathetus tan α = a ; tan β = b Affiliate b a

Exercise 4: Enter the letters on the sides so that the sine, cosine and tangent values ​​are correct.

Attempts: 0

→  Opposite cathete
hypotenuse

Exercise 5: Use the white slider to set the angle α of the triangle to the number of degrees in the table. Then transfer the values ​​you are looking for.

 sin α = a1 = = a2 = = c1 10 centimeters c2 The values ​​are rounded.

Attempts: 0

Exercise 6: Enter the letters on the sides so that the sine values ​​are correct.

Attempts: 0

Exercise 7: Enter the sine values ​​of the displayed angles in the text fields. Round to the fourth decimal place.

correct: 0 incorrect: 0

### Info: Calculate side lengths with the sine

The sine of an angle makes it possible for a right triangle to calculate the length of its opposite cathetus or the hypotenuse.

 a = sin α * c b = sin β * c

Exercise 8: Calculate the length of the red sides and enter them in the corresponding text field. Round to one decimal place.

Attempts: 0

Exercise 9: A triangle has the angles

Answer: The side is cm long. Round to the decimal place.

correct: 0 incorrect: 0

### Info: Calculate an angle in a right triangle using the aspect ratio of the opposite cathetus to the hypotenuse (sine).

If you divide the opposite side of an angle by the hypotenuse, you get its sine value. If this value is entered in the inverse function of the sine (arcsine), the size of the angle is obtained.

Example:

• a = 3 cm; c = 6 cm; γ = 90 °
•  3 = sin α = 0.5 6
• α = 30 ° (arcsine of 0.5)

Exercise 10: Enter the angles for the given sine values. Round to whole degrees.

correct: 0 incorrect: 0

Exercise 12: Find the angles α and β. Round to one decimal place.

Attempts: 0

Exercise 13: Enter the desired angle (α or β) of the right triangle with γ = 90 °. Round to whole degrees.

α = °

correct: 0 incorrect: 0

Exercise 14: How big is the slope angle α for the following skater ramp? Round to one decimal place.

Answer: The slope angle α is °

Attempts: 0

### Cosine problems

→  Affiliate
hypotenuse

Exercise 15: Enter the letters on the sides so that the cosines are correct.

Attempts: 0

Exercise 16: Enter the cosine values ​​of the displayed angles in the text fields. Round to the fourth decimal place.

correct: 0 incorrect: 0

### Info: Calculate side lengths with the cosine

The cosine of an angle makes it possible for a right triangle to calculate the length of its adjacent side or the hypotenuse.

 b = cos α * c a = cos β * c

Exercise 17: Calculate the length of the red sides and enter them in the corresponding text field. Round to one decimal place.

Attempts: 0

Exercise 18: A triangle has the angles

Answer: The side is cm long. Round to the decimal place.

correct: 0 incorrect: 0

### Info: Calculate an angle in a right triangle using the aspect ratio of adjacent to hypotenuse (cosine).

If you divide the adjacent side of an angle by the hypotenuse, you get its cosine value. If this value is entered into the inverse function of the cosine (arccosine), the size of the angle is obtained.

Example:

• b = 3 cm; c = 6 cm; γ = 90 °
•  3 = cos α = 0.5 6
• α = 60 ° (arccosine of 0.5)

Exercise 19: Enter angles for the given cosine values. Round to whole degrees.

correct: 0 incorrect: 0

Exercise 21: Find the angles α and β. Round to whole degrees.

Attempts: 0

Exercise 22: Enter the desired angle (α or β) of the right triangle with γ = 90 °. Round to whole degrees.

α = °

correct: 0 incorrect: 0

### Tangent problems

→  Opposite cathetus
Affiliate

Exercise 23: Enter the letters on the sides so that the tangents are correct.

Attempts: 0

Exercise 24: Enter the tangent values ​​of the angles shown in the text fields. Round to the fourth decimal place.

correct: 0 incorrect: 0

### Info: Calculate side lengths with the tangent

The tangent of an angle makes it possible for a right triangle to calculate the length of its opposite side or its adjacent side.

 a = tan α * b b = tan β a

Exercise 25: Calculate the length of the red sides and enter it in the corresponding text field. Round to one decimal place.

Attempts: 0

Exercise 26: A triangle has the angles

Answer: The side is cm long. Round to the decimal place.

correct: 0 incorrect: 0

### Info: Calculate an angle in a right-angled triangle with the help of the aspect ratio of the opposite side to the adjacent side (tangent).

If you divide the opposite side of an angle by its adjacent side, you get its tangent value. If this value is entered into the inverse function of the tangent (arctangent), the size of the angle is obtained.

Example:

• a = 5 cm; b = 5 cm; γ = 90 °
•  5 = tan α = 1 5
• α = 45 ° (arctangent of 1)

Exercise 27: Enter angles for the specified tangent values. Round to whole degrees.

correct: 0 incorrect: 0

Exercise 29: Find the angles α and β. Round to whole degrees.

Attempts: 0

Exercise 30: Enter the desired angle (α or β) of the right triangle with γ = 90 °. Round to whole degrees.

α = °

correct: 0 incorrect: 0

Exercise 31: A triangle has the angles

Answer: The side is cm long. Round to the decimal place.

correct: 0 incorrect: 0

Exercise 32: Enter the desired angle (α or β) of the right triangle with γ = 90 °. Round to whole degrees.

α = °

correct: 0 incorrect: 0

Exercise 33: Click on the values ​​that match the triangle in the angle functions.

 cos β = cos γ =

Attempts: 0

Exercise 34: A square pyramid is 220 m long. Its tip is aimed at 161 m from the bottom edge at an angle of 22 °. How tall is the pyramid? Round to one decimal place.

The pyramid has a height of m.

Attempts: 0

Exercise 35: The course of a soapbox race has a gradient of 18 ° over the first 40 meters. The following 116 meters to the finish line have a gradient of 8 °. What is the difference in altitude between the start and finish? Round to one decimal place.

The start of the race track is meters above the finish.

Attempts: 0

 Exercise 36: Two one-meter-wide walls stand parallel at a distance of 6.5 m from each other. One is 7.00 m, the other 4.08 m high. The carpenter is supposed to connect both walls with a sloping roof.a) At what angle (α) is the roof pitch to the supporting surfaces of the walls?b) How long is the roof slope?

Attempts: 0

Exercise 37: The valley station of a cable car is at a height of 1,258 m and the average angle of incline is α = 15 °. The steel cable has a length of 2.5 km. Enter the height of the mountain station. Round to whole meters.

The mountain station is at a height of m.

Attempts: 0

Exercise 38: The chord of a concrete cone has an angle of 65 ° to the base. The circumference of the cone is 3.78 m. What is its height? Round to inches.

The cone is meters high.

Attempts: 0

Exercise 39: Enter the length of the distance CD. Round to one decimal place.

The CD section is cm long.

Attempts: 0

 Exercise 40: a) Enter the length of the route CD. Round to two decimal places. b) Enter the angle β. Round to the full degree.

Attempts: 0

Exercise 41: The sides of a stepladder are 3 meters long. The opening angle is 42 °. What is the height of the ladder? Round to cm.

The ladder has a height of m.

Attempts: 0

Exercise 43: Enter the angle α of the trapezoid. Round to whole degrees. Then calculate the circumference (u) and enter the integer part of it.