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4.1 – Trigonometric Ratios


  • Define the sine, cosine, and tangent functions.
  • Calculate the sines, cosines, and tangents of the angles of special right triangles.
  • Use trigonometric ratios to solve real-world problems.


Key Terms

  • Cosine (COS) – In a right triangle, the ratio of the length of the angle’s adjacent leg to the length of the hypotenuse.
  • Sine (SIN) – In a right triangle, the ratio of the length of the angle’s opposite leg to the length of the hypotenuse.
  • Tangent (TAN) – In a right triangle, the ratio of the length of the angle’s opposite leg to the length of the adjacent leg.
  • Trigonometric Ratios – The ratios of the lengths of the sides of a right triangle. The three basic trigonometric ratios are sine, cosine, and tangent.





  • The hypotenuse is the side opposite the right angle.
  • The legs are the two sides opposite the acute angles.
  • In Special Right Triangles, we learned the formulas 30-60-90 and 45-45-90
    • See how the legs and hypotenuses relate to one another, below:

GeoB 4.1 30-60-90


GeoB 4.1 45-45-90



  • In trigonometry
    • θ (theta) is for unknown angles
    • x is for unknown sides


  • Labeling a Triangle Based on an Acute Angle
    • For angle θ (theta), notice which side is labeled as “opposite” and which side is labeled as “adjacent”
      • The side touching the angle is always adjacent
      • The side across from the angle is always opposite
      • The hypotenuse never changes.  It is always across from the 90° angle.

GeoA 4.1 Triangle Diagram


  • In this animation, notice which sides are Sine, Cosine, and Tangent from angle θ (theta).


  • Trigonometric Relationship Chart

GeoB 4.1 Trig Relationships

  • Trigonometric Functions of Special Right Triangles

GeoB 4.1 Trig 30hr-bar GeoB 4.1 Trig 45hr-bar GeoB 4.1 Trig 60


  • Trigonometric Functions Chart

GeoB 4.1 Trig Chart

  • Example 1:  What is the tangent of 67°?

GeoB 4.1 Tangent Ex

Answer: \frac{12}{5}


  • Example 2: What is the approximate value of x in the diagram below?
    • Hint: you will need to use one of the trigonometric ratios given in the box below.
    • Requirement: set your calculator to “degrees”

GeoB 4.1 Trig Missing Side

  • Solution:  Note that we do NOT have a hypotenuse.  We will need to use Tangent, since Tan=\frac{Opp}{Adj}
    • Tangent of 36° or Tangent of 54° will work.
      • Tan 36° is given in the box above (let’s use that)!
        • Tan 36° = 0.727
      • Tan 36° also equals \frac{23}{x}
      • Therefore, 0.727 must equal \frac{23}{x}
    • Solve for x by multiplying x on both sides, then dividing both sides by 0.727.
      • 0.727x = 23
      • x=\frac{23}{0.727}
    • Answer: 31.64


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