Lesson 6.4 Right Triangle Trigonometry

Finding Equivalent CoFunctions Ex. Find a cofunction equivalent to cos 2π/13. (or just use your table)

Any pair of trigonometric functions are cofunctions if the value of a trigonometric function of angle θ is equal to the cofunction of the complement of angle θ. Two angles are complementary if the sum of their measures is π/2 radians. The pair of acute angles in every right triangle is complementary.​ Thus, if θ is an acute angle of a right​ triangle, then the measure of the other acute angle is π/2−θ. Identify the cofunction identity for cos θ. cos θ=sin (π/2−θ) Calculate the cofunction of cos 2π/13. cos θ= sin (π/2−θ) cos 2π/13 =sin (π/2−2π/13) Substitute θ=2π/13. =sin (13π/26−4π/26) Rewrite using the least common denominator. =sin 9π/26 Subtract. ​ Hence, the cofunction of cos 2π/13 is sin 9π/26.

Which of the following is not a cofunction​ identity? Choose the correct choice below. A. tan θ=cot (π/2−θ) B. sec θ=cos (π/2−θ) C.cot θ=tan (π/2−θ) D.cos θ=sin (π/2−θ)

B. sec θ=cos (π/2−θ)

Using Special Right Triangles to Evaluate an Expression Ex. Use special right triangles to evaluate the expression. cot π/6 (or just use your table)

First recall the definition of the cotangent of an angle θ​, given a right triangle with acute angle θ and side lengths of​ hyp, opp, and adj. cotθ=adj/opp Now draw the special right triangle that includes the angle π/6​ (the π/6​, π/3​, π/2 triangle.) ​Finally, use the drawing of the π/6​, π/3​, π/2 triangle to find cot π/6. Notice that the side adjacent to the π/6 angle has length square root of 3 and the side opposite the π/6 angle has length 1. cot π/6=adj/opp = square root of 3/1 Substitute square root of 3 for adj and 1 for opp. = square root of 3 (Simplify.) ​ Therefore, cot π/6= square root of 3.

Rewriting Expressions as one of the 6 Trig Functions Ex. Rewrite the expression tan(90°−θ) secθ as one of the six trigonometric functions of acute angle θ.

To simplify the given​ expression, first rewrite the first factor so that its angle measure is θ using the cofunction identity cotθ=tan(90°−θ). tan(90°−θ) secθ= cotθ secθ Now write each factor in the expression in terms of sine and cosine. cotθ secθ=cos θ/sin θ 1/cos θ Notice that the common factor cosθ can be factored from the expression. cos θ sin θ1cos θ =1/sinθ =csc θ Simplify the expression. ​ Thus, tan(90°−θ) secθ= csc θ.

Determining Exact Values of Functions Ex. Determine the exact value of cos 2π/13 sin 9π/26+ sin 2π/13 cos 9π/26.

Notice that the angles in the trigonometric​ functions, 2π/13 and 9π/26​, are complements of each​ other; that​ is, their sum is π/2. Use cofunction identities to rewrite the terms in the given expression. Rewrite the first factor in the first term of the expression using the cofunction identities. Recall that the cofunction identity for cosθ is cosθ=sin (π/2−θ). Use the cofunction identity cosθ=sin (π/2−θ) to rewrite the first factor in the first term of the given expression. cos 2π/13 sin 9π/26+ sin 2π/13 cos 9π/26 ​=sin (π/2−2π/13) sin 9π/26+ sin 2π/13 cos 9π/26 ​=sin 9π/26 sin 9π/26+ sin 2π/13 cos 9π/26 Notice that the first term in the​ expression, sin 9π/26 sin 9π/26​, can be written as sin^2 9π26. sin 9π/26 sin 9π/26+ sin 2π/13 cos 9π/26 =sin^2 9π/26+ sin 2π/13 cos 9π/26 Now use the cofunction identity sinθ=cos (π/2−θ) to rewrite the first factor in the second term of this expression. sin^2 9π/26+ sin 2π/13 cos 9π/26 =sin^2 9π/26+ cos (π/2−2π/13) cos 9π/26 =sin^2 9π/26+ cos 9π/26 cos 9π/26 Observe that the second​ term, cos 9π/26 cos 9π/26​, can be written as cos^2 9π/26. sin^2 9π/26+ cos 9π/26 cos 9π/26 =sin^2 9π/26+ cos^2 9π/26 Next use the Pythagorean identity sin^2 θ+ cos^2 θ=1 to simplify the expression. sin^2 9π/26+ cos^2 9π/26=1 ​ Thus, cos 2π/13 sin 9π/26+ sin 2π/13 cos 9π/26=1.

Which of the following is not a fundamental​ identity? Choose the correct choice below. A. sec θ=1/cos θ B. sec^2 θ+1=tan^2 θ C. 1+cot^2 θ=csc^2 θ D. cot θ=cos θ/sin θ

B. sec^2 θ+1=tan^2 θ

Suppose that a right triangle has an acute angle θ and side lengths of​ hyp, opp, and adj. Which of the following does not accurately define a trigonometric​ function? Choose the equation that does not define a trigonometric function. A. secθ=hyp/adj B. sinθ=opp/hyp C. cscθ=hyp/opp D. cotθ=opp/adj

D. cotθ=opp/adj

complementary angles

Two positive angles are said to be complementary if the sum of their measures is π/2 radians (or 90°). Complementary angles always come in pairs. The acute angles of every right triangle are complementary.

What are the two special right triangles?

45-45-90 (π/4, π/4, π/2) and 30-60-90 (π/6, π/3, π/2)

Which of the following is not a fundamental​ identity? Choose the correct choice below. A. cot θ=1/tan θ B. 1=sin^2 θ+cos^2 θ C. 1+tan^2 θ=sec^2 θ D. cos θ=tan θ/sin θ.

D. cos θ=tan θ/sin θ.

Rewriting Trig Functions in terms of their cofunction Ex. Rewrite cot 81° in terms of its cofunction.

Each trigonometric function has a cofunction. The​ sine's cofunction is the​ cosine, the​ tangent's cofunction is the cotangent and the​ secant's cofunction is the cosecant.​ Thus, the abbreviation of the cofunction of cotθ is tanθ. According to the cofunction​ identities, the trigonometric function of any angle is equal to the cofunction of the​ angle's complement. ​ Therefore, cot 81°= tan 9°.

Simplifying An Expression Using A Common Denominator Ex. Simplify the expression by writing as a single term using a common denominator. π/2−3π/10

In order to find a common denominator, you must find the least common multiple (LCM) between the two numbers. In this case, the LCM of 2 and 10 would be 10. π/2− 3π/10 =5π/10− 3π/10 =2π/10 =π/5 (Simplify)

What are the Four Fundamental Trigonometric Identities?

Quotient, Reciprocal, Pythagorean, and Cofunction

Simplifying Radicals

Think of perfect squares that are factors

Which of the following is not a valid​ equation? Choose the correct choice below. A. sin π/4=cos π/4 B. csc π/6=cosπ/3 C. tan π/4=cot π/4 D. sin π/3=cos π/6

B. csc π/6=cosπ/3

Using Identities To Find Exact Values of A Given Expression Ex. Use identities to find the exact value of the trigonometric expression cot69°− cos69°/sin69°

Recall Your Trigonometric Function Identities Notice that the quotient identity cotθ=cosθ/sinθ can be used to simplify the second term in the given expression. ​ Thus, cos69°/sin69°=cot69°. Use the quotient identity for cotangent to simplify the given trigonometric expression. cot69°−cos69°/sin69°= cot69°−cot69° Rewrite cos69°/sin69° using a quotient identity. =0 (Simplify.) ​ Thus, cot69°−cos69°/sin69°=0.

Evaluating the 6 Trig Functions of A Right Triangle Ex. A right triangle with acute angle θ is given. Evaluate the six trigonometric functions of the acute angle θ. A right triangle has a vertical right leg labeled "36," a horizontal bottom leg labeled "15" with an opposite angle "theta," and a hypotenuse rising from the left to right.

The lengths of two sides of the triangle are given. The length of the side opposite angle θ is 15 and that of the side adjacent to angle θ is 36. ​First, determine the length of the hypotenuse. To find the length of the​ hypotenuse, use the Pythagorean​ Theorem, which states that (hyp)^2=(adj)^2+(opp)^2. A right triangle has a vertical right leg labeled "adj equals 36," a horizontal bottom leg labeled "opp equals 15" with an opposite angle "theta," and a hypotenuse rising from the left to right labeled "hyp." opp = 15 hyp= θ adj = 36 Use the Pythagorean Theorem. Substitute the side lengths for adj and​ opp, and simplify the right side. (hyp)^2=(adj)^2+(opp)^2 =362+152 =1521 Take the square root of both sides. (hyp)^2=1521 hyp=±39 Because hyp represents the length of a side of a​ triangle, use only the positive square root value.​ Therefore, hyp=39. Given a right triangle with acute angle θ and side lengths of ​hyp, opp​, and adj​, the formula for sinθ is sinθ=opp/hyp. Now find the sine of the angle θ. Recall that sinθ=opp/hyp. sinθ=15/39 =5/13 (Simplify.) Given a right triangle with acute angle θ and side lengths of ​hyp, opp​, and adj​, the formula for cosθ is cosθ=adj/hyp. Now find the cosine of the angle θ. Recall that cosθ=adj/hyp. cosθ=36/39 =12/13 (Simplify.) Given a right triangle with acute angle θ and side lengths of ​hyp, opp​, and adj​, the formula for tanθ is tanθ=opp/adj. Now find the tangent of the angle θ. Recall that tanθ=opp/adj. tanθ=15/36 =5/12 (Simplify.) Given a right triangle with acute angle θ and side lengths of ​hyp, opp​, and adj​, the formula for cscθ is cscθ=hyp/opp. Now find the cosecant of the angle θ. Recall that cscθ=hyp/opp. cscθ=39/15 =13/5 (Simplify.) Given a right triangle with acute angle θ and side lengths of ​hyp, opp​, and adj​, the formula for secθ is secθ=hyp/adj. Now find the secant of the angle θ. Recall that secθ=hyp/adj. secθ=39/36 =13/12 (Simplify.) Given a right triangle with acute angle θ and side lengths of ​hyp, opp​, and adj​, the formula for cotθ is cotθ=adj/opp. Now find the cotangent of the angle θ. Recall that cotθ=adj/opp. cotθ=36/15 =12/5 (Simplify.)

Determining the Measure of An Acute Angle Ex. Determine the measure of the acute angle θ for which cosθ= square root of 3/2. (or just use your table)

To determine the measure of the acute angle θ for which cosθ= square root of 3/2​, first define the cosecant of an acute angle of a right triangle. The cosecant of an acute angle of a right triangle is equal to the ratio of the length of the hypotenuse to the length of the side opposite the angle. Since cosθ=adj./hyp.= square root of 3/2. The triangle that is consistent with this information is as shown in the figure. hyp=2 adj= square root of 3 opp= ? To find the length of side adjacent the​ angle, θ​, use the Pythagorean theorem. (adj)2+(opp)2=(hyp)^2 = square root of 3^2+(opp)^2=(2​)2 Substitute square root of 3 for adj and 2 for hyp. 3+(opp)^2=4 Square on both the sides. Subtract 3 from 4. 3+(opp)^2=4 (opp)^2=1 Take the principal square root of each side and simplify. (opp)^2=1 opp=1 Now that the adjacent side has been​ found, determine whether the triangle is a special right triangle. This is a π6, π3, π2 special right triangle. Label the angles of the triangle. The measure of the acute angle θ for which cosθ= square root of 3/2 is as shown below. θ=π/6

Using Identities To Rewrite Expressions as Cos, Sin, Tan, Csc, Sec, or Cot Ex. Use identities to rewrite the following expression as sinθ​, cosθ​, tanθ​, cscθ​, secθ​, or cotθ. sin θ cot^2 θ sec θ

To express the given trigonometric expression as one of the 6 trigonometric​ functions, first write each factor of the expression in terms of sine​ and/or cosine. Notice that the quotient identity of cotangent allows cotangent to be written in terms of sine and cosine. The quotient identity of the cotangent function is shown below. cot θ=cos θ/sin θ The reciprocal identity allows secant to be written in terms of cosine. sec θ=1/cos θ Use the reciprocal and quotient identities to write the given expression in terms of sine and cosine. sin θcot^2 θsec θ =sin θ• cos^2 θ/sin^2 θ• 1cos θ Remove common factors from the expression and simplify. sin θ• cos^2θ/sin^2θ• 1cosθ =cos θ/sin θ =cot θ ​ Thus, sin θ cot^2 θsec θ= cot θ.

Finding the Remaining Trig Functions From A Given Function Ex. If θ is an acute angle of a right triangle and if cscθ= square root of 3​, then find the values of the remaining five trigonometric functions for angle θ.

You are given cscθ=hyp/opp= square root of 3​, where hyp= square root of 3 and opp=1. To find the length of the missing​ side, use the Pythagorean theorem. (adj)^2+(opp)^2=(hyp)^2 Substitute square root of 3 for hyp and 1 for opp. (adj)^2+1^2=​square root of (3​)^2 Square 3 and 1. (adj)^2+1=3 Subtract. ​(adj​)^2=2 Take the principal square root of each side and simplify. adj= square root of 2 Substitute the lengths of the appropriate sides into the given trigonometric ratio. Find the value of secθ. secθ=hyp/adj = square root of3/ square root of 2 Find the value of cotθ. cotθ=adj/opp = square root of 2 Find the value of sinθ. sinθ=opp/hyp =1/ square root of 3 Find the value of cosθ. cosθ=adj/hyp = square root of 2/ square root of 3 Find the value of tanθ. tanθ=opp/adj =1/ square root of 2