Properties of Right Triangles - SAT Math

1. Find the longer leg in a 30-60-90 triangle with shorter leg $3$.

   $3 \sqrt{3}$. Longer leg equals shorter leg times $\sqrt{3}$.

2. Find the hypotenuse if each leg is $7$ in a 45-45-90 triangle.

   $7\text{√}2$. Hypotenuse equals leg times $\sqrt{2}$ in 45-45-90 triangle.

3. Find $\text{tan } A$ if $\text{sin } A = 0.8$ and $\text{cos } A = 0.6$.

   $\text{tan } A = \frac{4}{3}$. Tangent equals sine divided by cosine.

4. State the formula for the hypotenuse in a right triangle with legs $a$ and $b$.

   $c = \sqrt{a^2 + b^2}$. Square root of sum of squared legs gives hypotenuse.

5. If $\text{sin } A = 0.6$, find $\text{cos } A$ in a right triangle.

   $\text{cos } A = 0.8$. Using $\sin^2 A + \cos^2 A = 1$ identity.

6. What is the cosine of angle $A$ in a right triangle?

   $\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of side adjacent to angle to hypotenuse.

7. What is the area formula for a right triangle with base $b$ and height $h$?

   $\frac{1}{2} \times b \times h$. Standard area formula for triangles using base and height.

8. State the sine function definition in a right triangle.

   $\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Sine relates the side opposite to the angle with the hypotenuse.

9. Find the length of the shorter leg in a 30-60-90 triangle with hypotenuse $10$.

   $5$. Shorter leg is half the hypotenuse in 30-60-90 triangle.

10. Find side $a$ if $b = 6$, $c = 10$ in a right triangle.

    $a = 8$. Using $a^2 + 6^2 = 10^2$, so $a^2 = 64$.

11. Calculate the tangent of angle $\theta$ given opposite = 5, adjacent = 12.

    $\tan(\theta) = \frac{5}{12}$. Tangent equals opposite divided by adjacent: $\frac{5}{12}$.

12. What is the sine of angle $A$ in a right triangle?

    $\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of side opposite angle to longest side.

13. What theorem states that $a^2 + b^2 = c^2$ in a right triangle?

    Pythagorean Theorem. Fundamental relationship between sides in right triangles.

14. What is the reciprocal of $\text{cos } A$?

    Secant. Secant is the reciprocal of cosine function.

15. Which trigonometric function equals $\frac{1}{2}$ at 30 degrees?

    Sine. In a 30-60-90 triangle, $\sin(30°) = \frac{1}{2}$.

16. Which trigonometric function is equal to $\frac{1}{\cos A}$?

    Secant. Reciprocal trigonometric function of cosine.

17. Identify the hypotenuse in a right triangle with sides 3, 4, and 5.

    5. The hypotenuse is always the longest side, opposite the right angle.

18. What is the formula for the area of a right triangle?

    $\frac{1}{2} \times \text{base} \times \text{height}$. Uses the two perpendicular sides as base and height.

19. Which trigonometric function is equal to $\frac{1}{\sin A}$?

    Cosecant. Reciprocal trigonometric function of sine.

20. What is the ratio of sides in a 45-45-90 triangle?

    $1:1:\text{√}2$. Special ratio for isosceles right triangle.

21. What is the value of $\text{sin } 30^\text{o}$?

    $\frac{1}{2}$. Standard trigonometric value for 30-degree angle.

22. What is the cosine function definition in a right triangle?

    $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. Cosine relates the side adjacent to the angle with the hypotenuse.

23. What is the value of $\text{sec } 0^\text{o}$?

    1. Secant of 0 degrees equals 1 divided by cos 0.

24. Identify the angle opposite the longest side in a right triangle.

    90 degrees. Right angle is always opposite the hypotenuse.

25. State the relationship between the angles in a right triangle.

    Sum is $180^\text{o}$. All triangle angles sum to 180 degrees.

26. Find the length of the hypotenuse in a right triangle with legs $5$ and $12$.

    $c = 13$. Using $5^2 + 12^2 = 25 + 144 = 169$.

27. What is the value of $\text{cot } 45^\text{o}$?

    1. Cotangent of 45 degrees equals cos 45 divided by sin 45.

28. What is the value of $\text{cos } 60^\text{o}$?

    $\frac{1}{2}$. Standard trigonometric value for 60-degree angle.

29. What is the reciprocal of $\text{tan } A$?

    Cotangent. Cotangent is the reciprocal of tangent function.

30. If $\text{tan } A = 1$, what is angle $A$?

    $45^\text{o}$. Angle whose tangent equals 1 is 45 degrees.

31. Find the missing side: $b = 8$, $c = 10$ in a right triangle.

    $a = 6$. Using $a^2 + 8^2 = 10^2$, so $a^2 = 36$.

32. What is the side ratio for a 30-60-90 triangle?

    $1 : \sqrt{3} : 2$. Standard side ratio for 30-60-90 special right triangle.

33. Identify the complementary angle to 60 degrees in a right triangle.

    30 degrees. Complementary angles in a right triangle sum to 90 degrees.

34. Find the missing leg if $a = 9$, $c = 15$ in a right triangle.

    $b = 12$. Using $9^2 + b^2 = 15^2$, so $b^2 = 144$.

35. What is the value of $\text{csc } 90^\text{o}$?

    1. Cosecant of 90 degrees equals 1 divided by sin 90.

36. What is the relation of angles in a 45-45-90 triangle?

    Each angle is $45^\text{o}$. Isosceles right triangle has two equal acute angles.

37. Calculate the tangent of a 45-degree angle in a right triangle.

    1. In a 45-45-90 triangle, opposite and adjacent sides are equal.

38. What is the value of $\text{sin } 90^\text{o}$?

    1. Sine of 90 degrees equals 1.

39. State the tangent function definition in a right triangle.

    $\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of opposite side to adjacent side.

40. Which trigonometric function is equal to $\frac{1}{\text{tan } A}$?

    Cotangent. Reciprocal trigonometric function of tangent.

41. Calculate the hypotenuse: $a = 9$, $b = 12$.

    $c = 15$. Using $9^2 + 12^2 = 81 + 144 = 225$.

42. What is the tangent of angle $A$ in a right triangle?

    $\frac{\text{opposite}}{\text{adjacent}}$. Ratio of opposite side to adjacent side.

43. Find the length of the hypotenuse if legs are 6 and 8.

    10. Apply Pythagorean theorem: $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

44. Which trigonometric function is equal to $\frac{1}{\text{sin } A}$?

    Cosecant. Reciprocal trigonometric function of sine.

45. Find the length of the hypotenuse: $a = 3$, $b = 4$.

    $c = 5$. Using $3^2 + 4^2 = 9 + 16 = 25$, so $c = 5$.

46. Find the other side if one leg is $5$ in a 45-45-90 triangle.

    $5$. Both legs are equal in isosceles right triangle.

47. Which trigonometric function is equal to $\frac{1}{\text{cos } A}$?

    Secant. Reciprocal trigonometric function of cosine.

48. What is the value of $\text{tan } 45^\text{o}$?

    1. Tangent equals 1 when opposite equals adjacent.

49. What is the value of $\text{cos } 0^\text{o}$?

    1. Cosine of 0 degrees equals 1.

50. What is the Pythagorean Theorem formula for a right triangle?

    $a^2 + b^2 = c^2$. The fundamental relationship between the three sides of a right triangle.

51. What is the reciprocal of $\text{sin } A$?

    Cosecant. Cosecant is the reciprocal of sine function.

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