Right Triangle Trigonometry Algebra III, Sec. 4.3 Objective Evaluate trigonometric functions of acute angles; Use the fundamental trigonometric identities.

The Six Trigonometric Functions θ opposite adjacent hypotenuse

The Six Trigonometric Functions The cosecant (csc) function is the reciprocal of the ______________ function. The cotangent (cot) function is the reciprocal of the _____________ function. The secant (sec) function is the reciprocal of the _____________ function. sine cosine tangent

Example 1 Find the values of the six trigonometric functions of θ. θ 2 5 First, find the missing side… Then, find the six trig fns…

Example 2 Find the values of cot 45° and csc 45°. 1, √2

Example 3 Use the equilateral triangle shown to find the values of cot 60° and cot 30°. √3/3, √3 30° 60° 2 √3 1

Example (on your handout) In the right triangle below, find sinθ, cosθ, and tanθ. θ 12 5

Cofunctions Cofunctions of complementary angles are _______. equal

Example 4 Use a calculator to evaluate… cot 34° 30’ 26”

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Example 5 Let θ be an acute angle such that cosθ = Find the values of (a) sinθ and (b) tanθ, using trigonometric identities. (a) 0.28 (b)

Example 6 Let β be an acute angle such that tanβ= 4. Find the values of (a) cotβ and (b) secβ, using trigonometric identities. (a) ¼ (b) √17

Example 7 Use trigonometric identities to transform the left side of the equation into the right side (0 < θ < π/2). a.

Example 7 Use trigonometric identities to transform the left side of the equation into the right side (0 < θ < π/2). b.

Applications  What does it mean to solve a right triangle?  Find all of the sides and angles!  The term angle of elevation means… the angle from the horizontal upward to an object.  The term angle of depression means… the angle from the horizontal downward to an object.

Applications: EXAMPLE 1  Solve ΔXYZ, given

ANSWER 1 Complementary angles

Applications: EXAMPLE 2  Solve ΔXYZ, given

 Hint: Use Calculator to change to a decimal. Then use inverse key. You always use the inverse key to find angle.

Applications: EXAMPLE 3 A surveyor found that the angle of elevation of the top of a flagpole was. The observation was made from a point 1.5 m above ground and 10 m from the base of the flagpole. Find the height of the flagpole to the nearest tenth of a meter.

ANSWER 3

The angle of depression from the top of a cliff 800 m high to the base of a log cabin is. How far is the cabin from the foot of the cliff? Applications: EXAMPLE 4

Alternate interior angles are congruent

Practice! Page 286 #